4 ESO
Educàlia editorial Mathem
atics 4 ESO Rosario Carrasco Torres
MATHEMATICS
Rosario Carrasco Torres
CUB_MATHEMATICS_4.indd 2-3 23/08/2019 11:32:09
MATHEMATICS 4 ESO
Rosario Carrasco Torres
tripas_MATHEMATICS_4.indd 1 23/08/2019 11:30:31
Primera edición 2019
Autora: Rosario Carrasco Torres
Maquetación: Rosario Carrasco Torres
Imprime: Grupo Digital 82, S.L.
ISBN: 978-84-17734-62-6
Depósito Legal: V-1444-2019
Printed in Spain/Impreso en España.
Todos los derechos reservados. No está permitida la reimpresión de ninguna parte de este libro, ni de imágenes ni de texto, ni tampoco su reproducción, ni utilización, en cualquier forma o por cualquier medio, bien sea electrónico, mecánico o de otro modo, tanto conocida como los que puedan inventarse, incluyendo el fotocopiado o grabación, ni está permitido almacenarlo en un sistema de información y recuperación, sin el permiso anticipado y por escrito del editor.
Alguna de las imágenes que incluye este libro son reproducciones que se han realizado acogiéndose al derecho de cita que aparece en el artículo 32 de la Ley 22/18987, del 11 de noviembre, de la Propiedad intelectual. Educàlia Editorial agradece a todas las instituciones, tanto públicas como privadas, citadas en estas páginas, su colaboración y pide disculpas por la posible omisión involuntaria de algunas de ellas.
Educàlia Editorial S.L.Avda. de las Jacarandas nº 2, loft 327 - 46100 Burjassot (València)960 624 309 - 963 768 542 - 610 900 [email protected]
"Thank you, Sue, I am privileged to have you as a friend.
I would also like to thank my dear students (2015-2019) for
encouraging me to write this book."
tripas_MATHEMATICS_4.indd 3 15/07/2019 10:47:36
tripas_MATHEMATICS_4.indd 2 23/08/2019 11:30:31
MATHEMATICS 4 ESO CONTENTS
NUMBERS AND ALGEBRA
GEOMETRY
STATISTICS AND PROBABILITY
page 15page 19page 67page 91page 120
page 140page 176page 212page 248page 287
page 313page 351page 379
tripas_MATHEMATICS_4.indd 7 23/08/2019 11:30:32
UNIT 5:
“PERIMETERS, SURFACES AND VOLUMES.
SIMILARITIES”
UNIT 5 PERIMETERS, SURFACES AND VOLUMES. SIMILARITIES 4 ESO
KEY VOCABULARY:
Polygon Regular Irregular Perimeter Area Surface Quadrilateral Composite Apothem Circular Sector Segment Annulus Polyhedron Platonic solid Prism Cuboid Convex Concave Pyramid Apex Surface development Tetrahedron Solid of revolution Axis of revolution Cylinder Cone Sphere Spherical cap Spherical wedge Ungula Similar polygons Ratio of similarity Ratio of proportionality
1. Polygons. Perimeter of a polygon
perimeter
“ :”
In the case of a regular polygonperimeter P
l
PERIMETERS, SURFACES AND VOLUMES. SIMILARITIES 4 ESO UNIT 5
In this unit you will review how to:
• Calculate the perimeter and the area of a polygon• Calculate the perimeter and the area of a circular shape• Calculate the area of a prism• Calculate the area of a pyramid• Calculate the area of a polyhedron• Calculate the area of a solid of revolution• Calculate the volume of a 3-D shape
In this unit you will learn how to: • Identify similar shapes• Calculate the ratio of similarity of similar shapes• Calculate the ratio of proportionality between areas of similar shapes• Calculate the ratio of proportionality between volumes of similar shapes
P = n · l
the base by using the Pythagoras’ theorem:
22 2426 −
140
UNIT 5:
“PERIMETERS, SURFACES AND VOLUMES.
SIMILARITIES”
UNIT 5 PERIMETERS, SURFACES AND VOLUMES. SIMILARITIES 4 ESO
KEY VOCABULARY:
Polygon Regular Irregular Perimeter Area Surface Quadrilateral Composite Apothem Circular Sector Segment Annulus Polyhedron Platonic solid Prism Cuboid Convex Concave Pyramid Apex Surface development Tetrahedron Solid of revolution Axis of revolution Cylinder Cone Sphere Spherical cap Spherical wedge Ungula Similar polygons Ratio of similarity Ratio of proportionality
page
tripas_MATHEMATICS_4.indd 140 23/08/2019 11:31:56
1. Polygons. Perimeter of a polygon
perimeter
“ :”
In the case of a regular polygonperimeter P
l
PERIMETERS, SURFACES AND VOLUMES. SIMILARITIES 4 ESO UNIT 5
In this unit you will review how to:
• Calculate the perimeter and the area of a polygon• Calculate the perimeter and the area of a circular shape• Calculate the area of a prism• Calculate the area of a pyramid• Calculate the area of a polyhedron• Calculate the area of a solid of revolution• Calculate the volume of a 3-D shape
In this unit you will learn how to: • Identify similar shapes• Calculate the ratio of similarity of similar shapes• Calculate the ratio of proportionality between areas of similar shapes• Calculate the ratio of proportionality between volumes of similar shapes
P = n · l
the base by using the Pythagoras’ theorem:
22 2426 −
141
1. Polygons. Perimeter of a polygon
perimeter
“ :”
In the case of a regular polygonperimeter P
l
PERIMETERS, SURFACES AND VOLUMES. SIMILARITIES 4 ESO UNIT 5
In this unit you will review how to:
• Calculate the perimeter and the area of a polygon• Calculate the perimeter and the area of a circular shape• Calculate the area of a prism• Calculate the area of a pyramid• Calculate the area of a polyhedron• Calculate the area of a solid of revolution• Calculate the volume of a 3-D shape
In this unit you will learn how to: • Identify similar shapes• Calculate the ratio of similarity of similar shapes• Calculate the ratio of proportionality between areas of similar shapes• Calculate the ratio of proportionality between volumes of similar shapes
P = n · l
the base by using the Pythagoras’ theorem:
22 2426 −
tripas_MATHEMATICS_4.indd 141 23/08/2019 11:31:57
UNIT 5:
“PERIMETERS, SURFACES AND VOLUMES.
SIMILARITIES”
UNIT 5 PERIMETERS, SURFACES AND VOLUMES. SIMILARITIES 4 ESO
KEY VOCABULARY:
Polygon Regular Irregular Perimeter Area Surface Quadrilateral Composite Apothem Circular Sector Segment Annulus Polyhedron Platonic solid Prism Cuboid Convex Concave Pyramid Apex Surface development Tetrahedron Solid of revolution Axis of revolution Cylinder Cone Sphere Spherical cap Spherical wedge Ungula Similar polygons Ratio of similarity Ratio of proportionality
1. Polygons. Perimeter of a polygon
perimeter
“ :”
In the case of a regular polygonperimeter P
l
PERIMETERS, SURFACES AND VOLUMES. SIMILARITIES 4 ESO UNIT 5
In this unit you will review how to:
• Calculate the perimeter and the area of a polygon• Calculate the perimeter and the area of a circular shape• Calculate the area of a prism• Calculate the area of a pyramid• Calculate the area of a polyhedron• Calculate the area of a solid of revolution• Calculate the volume of a 3-D shape
In this unit you will learn how to: • Identify similar shapes• Calculate the ratio of similarity of similar shapes• Calculate the ratio of proportionality between areas of similar shapes• Calculate the ratio of proportionality between volumes of similar shapes
P = n · l
the base by using the Pythagoras’ theorem:
22 2426 −
142
“ following regular polygon:”
2. Area of a polygonThe mathematical term “ ” or “ ” of a two
ONE TIP: Remember
those formulae
=A==A
=A
=A +=A
Long diagonal Short diagonal
Long base
Short base
tripas_MATHEMATICS_4.indd 142 23/08/2019 11:31:58
1. Polygons. Perimeter of a polygon
perimeter
“ :”
In the case of a regular polygonperimeter P
l
PERIMETERS, SURFACES AND VOLUMES. SIMILARITIES 4 ESO UNIT 5
In this unit you will review how to:
• Calculate the perimeter and the area of a polygon• Calculate the perimeter and the area of a circular shape• Calculate the area of a prism• Calculate the area of a pyramid• Calculate the area of a polyhedron• Calculate the area of a solid of revolution• Calculate the volume of a 3-D shape
In this unit you will learn how to: • Identify similar shapes• Calculate the ratio of similarity of similar shapes• Calculate the ratio of proportionality between areas of similar shapes• Calculate the ratio of proportionality between volumes of similar shapes
P = n · l
the base by using the Pythagoras’ theorem:
22 2426 −
143
In the case of a composite shape A
“Calculate the area of the following shapes:”
2
6 · π 2≈ 165,55cm
28· π·
4
3
2
21 · 2010·
2
7 27 +≈
tripas_MATHEMATICS_4.indd 143 23/08/2019 11:31:59
UNIT 5:
“PERIMETERS, SURFACES AND VOLUMES.
SIMILARITIES”
UNIT 5 PERIMETERS, SURFACES AND VOLUMES. SIMILARITIES 4 ESO
KEY VOCABULARY:
Polygon Regular Irregular Perimeter Area Surface Quadrilateral Composite Apothem Circular Sector Segment Annulus Polyhedron Platonic solid Prism Cuboid Convex Concave Pyramid Apex Surface development Tetrahedron Solid of revolution Axis of revolution Cylinder Cone Sphere Spherical cap Spherical wedge Ungula Similar polygons Ratio of similarity Ratio of proportionality
1. Polygons. Perimeter of a polygon
perimeter
“ :”
In the case of a regular polygonperimeter P
l
PERIMETERS, SURFACES AND VOLUMES. SIMILARITIES 4 ESO UNIT 5
In this unit you will review how to:
• Calculate the perimeter and the area of a polygon• Calculate the perimeter and the area of a circular shape• Calculate the area of a prism• Calculate the area of a pyramid• Calculate the area of a polyhedron• Calculate the area of a solid of revolution• Calculate the volume of a 3-D shape
In this unit you will learn how to: • Identify similar shapes• Calculate the ratio of similarity of similar shapes• Calculate the ratio of proportionality between areas of similar shapes• Calculate the ratio of proportionality between volumes of similar shapes
P = n · l
the base by using the Pythagoras’ theorem:
22 2426 −
144
In the case of a regular polygon with n sides A
Apolygon22
·side · nheight ·side=
“Calculate the area of the following shape:”
Aregular polygon =2
apothem ·perimeter
2
4 · 30
n is the number of triangles Notice that’s the perimeter of the polygon
: a hexagon can be divided into 6 triangles
apothem
apothem
Remember also that a hexagon is the only polygon for which the side
and the radius are same length.
925 −
tripas_MATHEMATICS_4.indd 144 23/08/2019 11:31:59
1. Polygons. Perimeter of a polygon
perimeter
“ :”
In the case of a regular polygonperimeter P
l
PERIMETERS, SURFACES AND VOLUMES. SIMILARITIES 4 ESO UNIT 5
In this unit you will review how to:
• Calculate the perimeter and the area of a polygon• Calculate the perimeter and the area of a circular shape• Calculate the area of a prism• Calculate the area of a pyramid• Calculate the area of a polyhedron• Calculate the area of a solid of revolution• Calculate the volume of a 3-D shape
In this unit you will learn how to: • Identify similar shapes• Calculate the ratio of similarity of similar shapes• Calculate the ratio of proportionality between areas of similar shapes• Calculate the ratio of proportionality between volumes of similar shapes
P = n · l
the base by using the Pythagoras’ theorem:
22 2426 −
145
3.Circular shapes.Perimeter and area of some circular shapes
Let’s study the
“
= 2πr
=
2r360
α ·r2·π·+
360
=
21PP · 2·π·
360+r
=21POPA
360αr·π ·2
triangle−
2πR + 2πr =
= 2π(R + r)
=
2·345 ·32·π·360
+ ≈
360
45 · π·3A2
= ≈ 3,53 cm
tripas_MATHEMATICS_4.indd 145 23/08/2019 11:32:00
UNIT 5:
“PERIMETERS, SURFACES AND VOLUMES.
SIMILARITIES”
UNIT 5 PERIMETERS, SURFACES AND VOLUMES. SIMILARITIES 4 ESO
KEY VOCABULARY:
Polygon Regular Irregular Perimeter Area Surface Quadrilateral Composite Apothem Circular Sector Segment Annulus Polyhedron Platonic solid Prism Cuboid Convex Concave Pyramid Apex Surface development Tetrahedron Solid of revolution Axis of revolution Cylinder Cone Sphere Spherical cap Spherical wedge Ungula Similar polygons Ratio of similarity Ratio of proportionality
1. Polygons. Perimeter of a polygon
perimeter
“ :”
In the case of a regular polygonperimeter P
l
PERIMETERS, SURFACES AND VOLUMES. SIMILARITIES 4 ESO UNIT 5
In this unit you will review how to:
• Calculate the perimeter and the area of a polygon• Calculate the perimeter and the area of a circular shape• Calculate the area of a prism• Calculate the area of a pyramid• Calculate the area of a polyhedron• Calculate the area of a solid of revolution• Calculate the volume of a 3-D shape
In this unit you will learn how to: • Identify similar shapes• Calculate the ratio of similarity of similar shapes• Calculate the ratio of proportionality between areas of similar shapes• Calculate the ratio of proportionality between volumes of similar shapes
P = n · l
the base by using the Pythagoras’ theorem:
22 2426 −
146
2725,47 · 45,9 22 − ≈
2
− 4,24 85 · π·7
360
2 ≈
2
1 · 2π(8 + 4) + 8 ≈
2
1 · π(8 – ) ≈ 75,4 cm
360140 · 10,64 · π· 2 + 20 ≈ 46 cm
21010,64 · 20 22 − ≈
360140 · π·10,64 2
– ≈ 101,96cm
tripas_MATHEMATICS_4.indd 146 23/08/2019 11:32:01
1. Polygons. Perimeter of a polygon
perimeter
“ :”
In the case of a regular polygonperimeter P
l
PERIMETERS, SURFACES AND VOLUMES. SIMILARITIES 4 ESO UNIT 5
In this unit you will review how to:
• Calculate the perimeter and the area of a polygon• Calculate the perimeter and the area of a circular shape• Calculate the area of a prism• Calculate the area of a pyramid• Calculate the area of a polyhedron• Calculate the area of a solid of revolution• Calculate the volume of a 3-D shape
In this unit you will learn how to: • Identify similar shapes• Calculate the ratio of similarity of similar shapes• Calculate the ratio of proportionality between areas of similar shapes• Calculate the ratio of proportionality between volumes of similar shapes
P = n · l
the base by using the Pythagoras’ theorem:
22 2426 −
147
4. Polyhedrons. Area of a polyhedron
that there are many types of polyhedrons
▪ The five
▪
TETRAHEDRON CUBE OCTAHEDRON DODECAHEDRON ICOSAHEDRON
1
2
BASE
Remember that’s a polyhedron called CUBOID, which is a Convex (a straight line can only cut its surface at two points) polyhedron
bounded by six rectangular faces.
LATERAL FACE
In this case, the polyhedron is Concave because a straight line can cut its surface at more than two points.
tripas_MATHEMATICS_4.indd 147 23/08/2019 11:32:01
UNIT 5:
“PERIMETERS, SURFACES AND VOLUMES.
SIMILARITIES”
UNIT 5 PERIMETERS, SURFACES AND VOLUMES. SIMILARITIES 4 ESO
KEY VOCABULARY:
Polygon Regular Irregular Perimeter Area Surface Quadrilateral Composite Apothem Circular Sector Segment Annulus Polyhedron Platonic solid Prism Cuboid Convex Concave Pyramid Apex Surface development Tetrahedron Solid of revolution Axis of revolution Cylinder Cone Sphere Spherical cap Spherical wedge Ungula Similar polygons Ratio of similarity Ratio of proportionality
1. Polygons. Perimeter of a polygon
perimeter
“ :”
In the case of a regular polygonperimeter P
l
PERIMETERS, SURFACES AND VOLUMES. SIMILARITIES 4 ESO UNIT 5
In this unit you will review how to:
• Calculate the perimeter and the area of a polygon• Calculate the perimeter and the area of a circular shape• Calculate the area of a prism• Calculate the area of a pyramid• Calculate the area of a polyhedron• Calculate the area of a solid of revolution• Calculate the volume of a 3-D shape
In this unit you will learn how to: • Identify similar shapes• Calculate the ratio of similarity of similar shapes• Calculate the ratio of proportionality between areas of similar shapes• Calculate the ratio of proportionality between volumes of similar shapes
P = n · l
the base by using the Pythagoras’ theorem:
22 2426 −
148
▪Let’s remember some
Let’s remember
Euler’s formula
F VE
tance, the Euler’s formula states for the following polyhedron despite
3
BASE
APEX
HEIGHT
Remember that, if their sides are regular polygons:
* The lateral apothem of a pyramid isa perpendicular line from the apexof the pyramid to a side of its base.
* The base apothem is a perpendicularline from the center of the base to aside of that base.BASE APOTHEM
LATERAL APOTHEM
F + V = E + 2
F = 7
V = 10
E = 15
tripas_MATHEMATICS_4.indd 148 23/08/2019 11:32:02
1. Polygons. Perimeter of a polygon
perimeter
“ :”
In the case of a regular polygonperimeter P
l
PERIMETERS, SURFACES AND VOLUMES. SIMILARITIES 4 ESO UNIT 5
In this unit you will review how to:
• Calculate the perimeter and the area of a polygon• Calculate the perimeter and the area of a circular shape• Calculate the area of a prism• Calculate the area of a pyramid• Calculate the area of a polyhedron• Calculate the area of a solid of revolution• Calculate the volume of a 3-D shape
In this unit you will learn how to: • Identify similar shapes• Calculate the ratio of similarity of similar shapes• Calculate the ratio of proportionality between areas of similar shapes• Calculate the ratio of proportionality between volumes of similar shapes
P = n · l
the base by using the Pythagoras’ theorem:
22 2426 −
149
“Calculate the area of the following surface development:”
236 · 6 22−
≈
210 · 6
≈
≈ 15,59 + 3 · 30 = 105,59 cm
Let’s see some formulas in order to calculate
4.1. Prisms. Area of a prism
Alateral = n · Arectangle
regular prism
: “ Calculate the area of the following prism : ”
224 · 4 · 6 22 −
≈ 41,57
≈ 6 · 4 · 12 + 2 · 41,57
A = Alateral + 2 · Abase
A = A lateral + 2 · Abase = Perimeterbase · height + 2 · Abase
tripas_MATHEMATICS_4.indd 149 23/08/2019 11:32:03
UNIT 5:
“PERIMETERS, SURFACES AND VOLUMES.
SIMILARITIES”
UNIT 5 PERIMETERS, SURFACES AND VOLUMES. SIMILARITIES 4 ESO
KEY VOCABULARY:
Polygon Regular Irregular Perimeter Area Surface Quadrilateral Composite Apothem Circular Sector Segment Annulus Polyhedron Platonic solid Prism Cuboid Convex Concave Pyramid Apex Surface development Tetrahedron Solid of revolution Axis of revolution Cylinder Cone Sphere Spherical cap Spherical wedge Ungula Similar polygons Ratio of similarity Ratio of proportionality
1. Polygons. Perimeter of a polygon
perimeter
“ :”
In the case of a regular polygonperimeter P
l
PERIMETERS, SURFACES AND VOLUMES. SIMILARITIES 4 ESO UNIT 5
In this unit you will review how to:
• Calculate the perimeter and the area of a polygon• Calculate the perimeter and the area of a circular shape• Calculate the area of a prism• Calculate the area of a pyramid• Calculate the area of a polyhedron• Calculate the area of a solid of revolution• Calculate the volume of a 3-D shape
In this unit you will learn how to: • Identify similar shapes• Calculate the ratio of similarity of similar shapes• Calculate the ratio of proportionality between areas of similar shapes• Calculate the ratio of proportionality between volumes of similar shapes
P = n · l
the base by using the Pythagoras’ theorem:
22 2426 −
150
2
22 821 · 16 · 8 − ≈
≈
4. 2. Pyramids. Area of a pyramid
Alateral = n · Atriangle
2
height · basetriangletriangle
2apothem · Perimeter base
regular pyramid
: “ Calculate the area of the following ”
. Let’s calculate its area:
2
22 48 · 8 · 3 −
2
22 48 · 8 − ≈
A = Alateral + Abase
A = A lateral + Abase = + Abase
Don’t forget that the height of the
tripas_MATHEMATICS_4.indd 150 23/08/2019 11:32:04
1. Polygons. Perimeter of a polygon
perimeter
“ :”
In the case of a regular polygonperimeter P
l
PERIMETERS, SURFACES AND VOLUMES. SIMILARITIES 4 ESO UNIT 5
In this unit you will review how to:
• Calculate the perimeter and the area of a polygon• Calculate the perimeter and the area of a circular shape• Calculate the area of a prism• Calculate the area of a pyramid• Calculate the area of a polyhedron• Calculate the area of a solid of revolution• Calculate the volume of a 3-D shape
In this unit you will learn how to: • Identify similar shapes• Calculate the ratio of similarity of similar shapes• Calculate the ratio of proportionality between areas of similar shapes• Calculate the ratio of proportionality between volumes of similar shapes
P = n · l
the base by using the Pythagoras’ theorem:
22 2426 −
151
4. 3. Area of a 3D-shape
irregular polyhedron
: “ Calculate the area of the following irregular pyramid: ”
Pythagoras’ theorem:
Let’s do the calculations:
22 1118 + ≈ 22 5,718 + ≈
≈221,1 · 22
219,5 · 15
composite 3D-shape
tripas_MATHEMATICS_4.indd 151 23/08/2019 11:32:04
UNIT 5:
“PERIMETERS, SURFACES AND VOLUMES.
SIMILARITIES”
UNIT 5 PERIMETERS, SURFACES AND VOLUMES. SIMILARITIES 4 ESO
KEY VOCABULARY:
Polygon Regular Irregular Perimeter Area Surface Quadrilateral Composite Apothem Circular Sector Segment Annulus Polyhedron Platonic solid Prism Cuboid Convex Concave Pyramid Apex Surface development Tetrahedron Solid of revolution Axis of revolution Cylinder Cone Sphere Spherical cap Spherical wedge Ungula Similar polygons Ratio of similarity Ratio of proportionality
1. Polygons. Perimeter of a polygon
perimeter
“ :”
In the case of a regular polygonperimeter P
l
PERIMETERS, SURFACES AND VOLUMES. SIMILARITIES 4 ESO UNIT 5
In this unit you will review how to:
• Calculate the perimeter and the area of a polygon• Calculate the perimeter and the area of a circular shape• Calculate the area of a prism• Calculate the area of a pyramid• Calculate the area of a polyhedron• Calculate the area of a solid of revolution• Calculate the volume of a 3-D shape
In this unit you will learn how to: • Identify similar shapes• Calculate the ratio of similarity of similar shapes• Calculate the ratio of proportionality between areas of similar shapes• Calculate the ratio of proportionality between volumes of similar shapes
P = n · l
the base by using the Pythagoras’ theorem:
22 2426 −
152
: “ Calculate the area of the following shape:”
2
22 710 · 14 −
2
22 310 · 6 −
≈
≈
5. Solids of revolution. Area of a solid of revolution
axis of revolution solid of revolution
tripas_MATHEMATICS_4.indd 152 23/08/2019 11:32:05
1. Polygons. Perimeter of a polygon
perimeter
“ :”
In the case of a regular polygonperimeter P
l
PERIMETERS, SURFACES AND VOLUMES. SIMILARITIES 4 ESO UNIT 5
In this unit you will review how to:
• Calculate the perimeter and the area of a polygon• Calculate the perimeter and the area of a circular shape• Calculate the area of a prism• Calculate the area of a pyramid• Calculate the area of a polyhedron• Calculate the area of a solid of revolution• Calculate the volume of a 3-D shape
In this unit you will learn how to: • Identify similar shapes• Calculate the ratio of similarity of similar shapes• Calculate the ratio of proportionality between areas of similar shapes• Calculate the ratio of proportionality between volumes of similar shapes
P = n · l
the base by using the Pythagoras’ theorem:
22 2426 −
153
90
Let’s
spherical shapes
The area of a cone is:
A = Alateral Abase = πr + πrA = πrg + πr Circular sector Circle
The area of a cylinder is: A = Alateral · Abase= 2πr · h + 2 · πr
The area of a spherical cap is : A πrThe area of a sphere is: A = 4πr
The area of a spherical segment is : A 2πrh
h
α
The area of a spherical wedge is : A =360
α4π 2r
tripas_MATHEMATICS_4.indd 153 23/08/2019 11:32:06
UNIT 5:
“PERIMETERS, SURFACES AND VOLUMES.
SIMILARITIES”
UNIT 5 PERIMETERS, SURFACES AND VOLUMES. SIMILARITIES 4 ESO
KEY VOCABULARY:
Polygon Regular Irregular Perimeter Area Surface Quadrilateral Composite Apothem Circular Sector Segment Annulus Polyhedron Platonic solid Prism Cuboid Convex Concave Pyramid Apex Surface development Tetrahedron Solid of revolution Axis of revolution Cylinder Cone Sphere Spherical cap Spherical wedge Ungula Similar polygons Ratio of similarity Ratio of proportionality
1. Polygons. Perimeter of a polygon
perimeter
“ :”
In the case of a regular polygonperimeter P
l
PERIMETERS, SURFACES AND VOLUMES. SIMILARITIES 4 ESO UNIT 5
In this unit you will review how to:
• Calculate the perimeter and the area of a polygon• Calculate the perimeter and the area of a circular shape• Calculate the area of a prism• Calculate the area of a pyramid• Calculate the area of a polyhedron• Calculate the area of a solid of revolution• Calculate the volume of a 3-D shape
In this unit you will learn how to: • Identify similar shapes• Calculate the ratio of similarity of similar shapes• Calculate the ratio of proportionality between areas of similar shapes• Calculate the ratio of proportionality between volumes of similar shapes
P = n · l
the base by using the Pythagoras’ theorem:
22 2426 −
154
: “Calculate the areas of the following shapes:”
=360
35 · 18 ·4π 2
≈
= 2 · 7 · π· 2 ≈ 87,96 cm
= 3,2 · 6 · π· 2 ≈
composite 3D-shapes that include shapes of revolution.
: “ shapes:”
A = Acylinder without bases + Asphere = 2π ·
− 224225 · 24 + 4π ·
2224225
− ≈
A = Acone without base + Asemi-sphere + Aprism 1 + Aprism 2 =
= π · 0,4 · 24,022 + 2
20,4 · 4π
– 2 · 1 + 2 · 2 · 1) ≈ 29,57 m
tripas_MATHEMATICS_4.indd 154 23/08/2019 11:32:07
1. Polygons. Perimeter of a polygon
perimeter
“ :”
In the case of a regular polygonperimeter P
l
PERIMETERS, SURFACES AND VOLUMES. SIMILARITIES 4 ESO UNIT 5
In this unit you will review how to:
• Calculate the perimeter and the area of a polygon• Calculate the perimeter and the area of a circular shape• Calculate the area of a prism• Calculate the area of a pyramid• Calculate the area of a polyhedron• Calculate the area of a solid of revolution• Calculate the volume of a 3-D shape
In this unit you will learn how to: • Identify similar shapes• Calculate the ratio of similarity of similar shapes• Calculate the ratio of proportionality between areas of similar shapes• Calculate the ratio of proportionality between volumes of similar shapes
P = n · l
the base by using the Pythagoras’ theorem:
22 2426 −
155
6. Volume of a 3D - shape
Let’s remember
A = πr2
V = Abase · h V = Abase · h = πr2 · h
V = 31
· Abase · hV =
31
· πr2 · h
V = 34
· πr3
There are some curiosities that may help you remember the previous formulae:
Vcone = 31
Vcylinder
Vsphere = 32
· Vcylinder
(The height of the cylinder is double the radius of the sphere)
tripas_MATHEMATICS_4.indd 155 23/08/2019 11:32:08
UNIT 5:
“PERIMETERS, SURFACES AND VOLUMES.
SIMILARITIES”
UNIT 5 PERIMETERS, SURFACES AND VOLUMES. SIMILARITIES 4 ESO
KEY VOCABULARY:
Polygon Regular Irregular Perimeter Area Surface Quadrilateral Composite Apothem Circular Sector Segment Annulus Polyhedron Platonic solid Prism Cuboid Convex Concave Pyramid Apex Surface development Tetrahedron Solid of revolution Axis of revolution Cylinder Cone Sphere Spherical cap Spherical wedge Ungula Similar polygons Ratio of similarity Ratio of proportionality
1. Polygons. Perimeter of a polygon
perimeter
“ :”
In the case of a regular polygonperimeter P
l
PERIMETERS, SURFACES AND VOLUMES. SIMILARITIES 4 ESO UNIT 5
In this unit you will review how to:
• Calculate the perimeter and the area of a polygon• Calculate the perimeter and the area of a circular shape• Calculate the area of a prism• Calculate the area of a pyramid• Calculate the area of a polyhedron• Calculate the area of a solid of revolution• Calculate the volume of a 3-D shape
In this unit you will learn how to: • Identify similar shapes• Calculate the ratio of similarity of similar shapes• Calculate the ratio of proportionality between areas of similar shapes• Calculate the ratio of proportionality between volumes of similar shapes
P = n · l
the base by using the Pythagoras’ theorem:
22 2426 −
156
93
Let’s take a look at the following shapes that illustrate the curiosities we have
: “Calculate the volumes of the following objects:”
V = 3
1Abase · h =
3
1
2
apothem· perimeter · h
3
1
2
3 · 4 · 5
V = Abase · h = πr2 · h π · 5
V = Vtriangular prism + Voctagonal prism = 242
816 · 16 · 22 −
242
821 · 16 · · 822 −
7. Similar polygons. Ratio of similarity
tripas_MATHEMATICS_4.indd 156 23/08/2019 11:32:09
1. Polygons. Perimeter of a polygon
perimeter
“ :”
In the case of a regular polygonperimeter P
l
PERIMETERS, SURFACES AND VOLUMES. SIMILARITIES 4 ESO UNIT 5
In this unit you will review how to:
• Calculate the perimeter and the area of a polygon• Calculate the perimeter and the area of a circular shape• Calculate the area of a prism• Calculate the area of a pyramid• Calculate the area of a polyhedron• Calculate the area of a solid of revolution• Calculate the volume of a 3-D shape
In this unit you will learn how to: • Identify similar shapes• Calculate the ratio of similarity of similar shapes• Calculate the ratio of proportionality between areas of similar shapes• Calculate the ratio of proportionality between volumes of similar shapes
P = n · l
the base by using the Pythagoras’ theorem:
22 2426 −
157
“Study if the following polygons are similar and if they are similar, calculate the ratio of similarity:”
5,16
9=
: “Given the following shape:
Which of the following shapes are similar to the given one?”
: “Given the following shapes, analyse which of them are similar and
corresponding angles
corresponding sides
tripas_MATHEMATICS_4.indd 157 23/08/2019 11:32:10
UNIT 5:
“PERIMETERS, SURFACES AND VOLUMES.
SIMILARITIES”
UNIT 5 PERIMETERS, SURFACES AND VOLUMES. SIMILARITIES 4 ESO
KEY VOCABULARY:
Polygon Regular Irregular Perimeter Area Surface Quadrilateral Composite Apothem Circular Sector Segment Annulus Polyhedron Platonic solid Prism Cuboid Convex Concave Pyramid Apex Surface development Tetrahedron Solid of revolution Axis of revolution Cylinder Cone Sphere Spherical cap Spherical wedge Ungula Similar polygons Ratio of similarity Ratio of proportionality
1. Polygons. Perimeter of a polygon
perimeter
“ :”
In the case of a regular polygonperimeter P
l
PERIMETERS, SURFACES AND VOLUMES. SIMILARITIES 4 ESO UNIT 5
In this unit you will review how to:
• Calculate the perimeter and the area of a polygon• Calculate the perimeter and the area of a circular shape• Calculate the area of a prism• Calculate the area of a pyramid• Calculate the area of a polyhedron• Calculate the area of a solid of revolution• Calculate the volume of a 3-D shape
In this unit you will learn how to: • Identify similar shapes• Calculate the ratio of similarity of similar shapes• Calculate the ratio of proportionality between areas of similar shapes• Calculate the ratio of proportionality between volumes of similar shapes
P = n · l
the base by using the Pythagoras’ theorem:
22 2426 −
158
m13
5,25,2
2
2=
15,0
5,225,1
2
1==
12
5,25
2
4==
: “Are similar?:”
43,5
6
3
Let’s remember how to construct similar polygons
“O expansion with ratio 3:”A B’ A’ B
C
C’ D
D’
E F’
F
E’
=
=
=
=
=
=
The ratio is 3 because for instance OA
OA'
That’s an Expansion. It is called a reduction when
the ratio is smaller than 1.
tripas_MATHEMATICS_4.indd 158 23/08/2019 11:32:11
1. Polygons. Perimeter of a polygon
perimeter
“ :”
In the case of a regular polygonperimeter P
l
PERIMETERS, SURFACES AND VOLUMES. SIMILARITIES 4 ESO UNIT 5
In this unit you will review how to:
• Calculate the perimeter and the area of a polygon• Calculate the perimeter and the area of a circular shape• Calculate the area of a prism• Calculate the area of a pyramid• Calculate the area of a polyhedron• Calculate the area of a solid of revolution• Calculate the volume of a 3-D shape
In this unit you will learn how to: • Identify similar shapes• Calculate the ratio of similarity of similar shapes• Calculate the ratio of proportionality between areas of similar shapes• Calculate the ratio of proportionality between volumes of similar shapes
P = n · l
the base by using the Pythagoras’ theorem:
22 2426 −
159
8. Ratio of proportionality of areas of similar shapes
“
one and the ratio of proportionality of their areas.”
3AD
D'A'= 3
4D'A'
= D'A'
3AB
B'A'= 3
12B'A'
= B'A'
48
432
the square
9. Ratio of proportionality of volumes of similar shapes
“Given the following similar .”
Let’s calculate their volumes:
tripas_MATHEMATICS_4.indd 159 23/08/2019 11:32:12
UNIT 5:
“PERIMETERS, SURFACES AND VOLUMES.
SIMILARITIES”
UNIT 5 PERIMETERS, SURFACES AND VOLUMES. SIMILARITIES 4 ESO
KEY VOCABULARY:
Polygon Regular Irregular Perimeter Area Surface Quadrilateral Composite Apothem Circular Sector Segment Annulus Polyhedron Platonic solid Prism Cuboid Convex Concave Pyramid Apex Surface development Tetrahedron Solid of revolution Axis of revolution Cylinder Cone Sphere Spherical cap Spherical wedge Ungula Similar polygons Ratio of similarity Ratio of proportionality
1. Polygons. Perimeter of a polygon
perimeter
“ :”
In the case of a regular polygonperimeter P
l
PERIMETERS, SURFACES AND VOLUMES. SIMILARITIES 4 ESO UNIT 5
In this unit you will review how to:
• Calculate the perimeter and the area of a polygon• Calculate the perimeter and the area of a circular shape• Calculate the area of a prism• Calculate the area of a pyramid• Calculate the area of a polyhedron• Calculate the area of a solid of revolution• Calculate the volume of a 3-D shape
In this unit you will learn how to: • Identify similar shapes• Calculate the ratio of similarity of similar shapes• Calculate the ratio of proportionality between areas of similar shapes• Calculate the ratio of proportionality between volumes of similar shapes
P = n · l
the base by using the Pythagoras’ theorem:
22 2426 −
160
40320 the cube
10. Similarity in real life
…
“
00000165000000018 ≈
3·9 · 4 3 ≈
r
tripas_MATHEMATICS_4.indd 160 23/08/2019 11:32:13
1. Polygons. Perimeter of a polygon
perimeter
“ :”
In the case of a regular polygonperimeter P
l
PERIMETERS, SURFACES AND VOLUMES. SIMILARITIES 4 ESO UNIT 5
In this unit you will review how to:
• Calculate the perimeter and the area of a polygon• Calculate the perimeter and the area of a circular shape• Calculate the area of a prism• Calculate the area of a pyramid• Calculate the area of a polyhedron• Calculate the area of a solid of revolution• Calculate the volume of a 3-D shape
In this unit you will learn how to: • Identify similar shapes• Calculate the ratio of similarity of similar shapes• Calculate the ratio of proportionality between areas of similar shapes• Calculate the ratio of proportionality between volumes of similar shapes
P = n · l
the base by using the Pythagoras’ theorem:
22 2426 −
161
32): 13-10 · (1,09 · · 4 3 ≈ 6,8 · 10
COMPLETE THE FOLLOWING TABLE :Key VOCABULARY phonetics meaning Polygon
Regular
Irregular
Perimeter
Area Surface Quadrilateral Composite Apothem Circular Sector Segment Annulus Polyhedron Platonic solid Prism Cuboid Convex Concave Pyramid Apex Surface development Tetrahedron Solid of revolution Axis of revolution Cylinder Cone Sphere Spherical cap Spherical wedge Ungula Similar Ratio of similarity Ratio of proportionality
tripas_MATHEMATICS_4.indd 161 23/08/2019 11:32:13
UNIT 5:
“PERIMETERS, SURFACES AND VOLUMES.
SIMILARITIES”
UNIT 5 PERIMETERS, SURFACES AND VOLUMES. SIMILARITIES 4 ESO
KEY VOCABULARY:
Polygon Regular Irregular Perimeter Area Surface Quadrilateral Composite Apothem Circular Sector Segment Annulus Polyhedron Platonic solid Prism Cuboid Convex Concave Pyramid Apex Surface development Tetrahedron Solid of revolution Axis of revolution Cylinder Cone Sphere Spherical cap Spherical wedge Ungula Similar polygons Ratio of similarity Ratio of proportionality
1. Polygons. Perimeter of a polygon
perimeter
“ :”
In the case of a regular polygonperimeter P
l
PERIMETERS, SURFACES AND VOLUMES. SIMILARITIES 4 ESO UNIT 5
In this unit you will review how to:
• Calculate the perimeter and the area of a polygon• Calculate the perimeter and the area of a circular shape• Calculate the area of a prism• Calculate the area of a pyramid• Calculate the area of a polyhedron• Calculate the area of a solid of revolution• Calculate the volume of a 3-D shape
In this unit you will learn how to: • Identify similar shapes• Calculate the ratio of similarity of similar shapes• Calculate the ratio of proportionality between areas of similar shapes• Calculate the ratio of proportionality between volumes of similar shapes
P = n · l
the base by using the Pythagoras’ theorem:
22 2426 −
162
EXERCISES
EXERCISES
25
UNIT 5 PERIMETERS, SURFACES AND VOLUMES. SIMILARITIES 4 ESOEXERCISES
tripas_MATHEMATICS_4.indd 162 23/08/2019 11:32:14
1. Polygons. Perimeter of a polygon
perimeter
“ :”
In the case of a regular polygonperimeter P
l
PERIMETERS, SURFACES AND VOLUMES. SIMILARITIES 4 ESO UNIT 5
In this unit you will review how to:
• Calculate the perimeter and the area of a polygon• Calculate the perimeter and the area of a circular shape• Calculate the area of a prism• Calculate the area of a pyramid• Calculate the area of a polyhedron• Calculate the area of a solid of revolution• Calculate the volume of a 3-D shape
In this unit you will learn how to: • Identify similar shapes• Calculate the ratio of similarity of similar shapes• Calculate the ratio of proportionality between areas of similar shapes• Calculate the ratio of proportionality between volumes of similar shapes
P = n · l
the base by using the Pythagoras’ theorem:
22 2426 −
163
EXERCISES
tripas_MATHEMATICS_4.indd 163 23/08/2019 11:32:15
UNIT 5:
“PERIMETERS, SURFACES AND VOLUMES.
SIMILARITIES”
UNIT 5 PERIMETERS, SURFACES AND VOLUMES. SIMILARITIES 4 ESO
KEY VOCABULARY:
Polygon Regular Irregular Perimeter Area Surface Quadrilateral Composite Apothem Circular Sector Segment Annulus Polyhedron Platonic solid Prism Cuboid Convex Concave Pyramid Apex Surface development Tetrahedron Solid of revolution Axis of revolution Cylinder Cone Sphere Spherical cap Spherical wedge Ungula Similar polygons Ratio of similarity Ratio of proportionality
1. Polygons. Perimeter of a polygon
perimeter
“ :”
In the case of a regular polygonperimeter P
l
PERIMETERS, SURFACES AND VOLUMES. SIMILARITIES 4 ESO UNIT 5
In this unit you will review how to:
• Calculate the perimeter and the area of a polygon• Calculate the perimeter and the area of a circular shape• Calculate the area of a prism• Calculate the area of a pyramid• Calculate the area of a polyhedron• Calculate the area of a solid of revolution• Calculate the volume of a 3-D shape
In this unit you will learn how to: • Identify similar shapes• Calculate the ratio of similarity of similar shapes• Calculate the ratio of proportionality between areas of similar shapes• Calculate the ratio of proportionality between volumes of similar shapes
P = n · l
the base by using the Pythagoras’ theorem:
22 2426 −
164
EXERCISES
tripas_MATHEMATICS_4.indd 164 23/08/2019 11:32:16
1. Polygons. Perimeter of a polygon
perimeter
“ :”
In the case of a regular polygonperimeter P
l
PERIMETERS, SURFACES AND VOLUMES. SIMILARITIES 4 ESO UNIT 5
In this unit you will review how to:
• Calculate the perimeter and the area of a polygon• Calculate the perimeter and the area of a circular shape• Calculate the area of a prism• Calculate the area of a pyramid• Calculate the area of a polyhedron• Calculate the area of a solid of revolution• Calculate the volume of a 3-D shape
In this unit you will learn how to: • Identify similar shapes• Calculate the ratio of similarity of similar shapes• Calculate the ratio of proportionality between areas of similar shapes• Calculate the ratio of proportionality between volumes of similar shapes
P = n · l
the base by using the Pythagoras’ theorem:
22 2426 −
165
EXERCISES
4,6cm 7,25cm
8,49cm
tripas_MATHEMATICS_4.indd 165 23/08/2019 11:32:16
UNIT 5:
“PERIMETERS, SURFACES AND VOLUMES.
SIMILARITIES”
UNIT 5 PERIMETERS, SURFACES AND VOLUMES. SIMILARITIES 4 ESO
KEY VOCABULARY:
Polygon Regular Irregular Perimeter Area Surface Quadrilateral Composite Apothem Circular Sector Segment Annulus Polyhedron Platonic solid Prism Cuboid Convex Concave Pyramid Apex Surface development Tetrahedron Solid of revolution Axis of revolution Cylinder Cone Sphere Spherical cap Spherical wedge Ungula Similar polygons Ratio of similarity Ratio of proportionality
1. Polygons. Perimeter of a polygon
perimeter
“ :”
In the case of a regular polygonperimeter P
l
PERIMETERS, SURFACES AND VOLUMES. SIMILARITIES 4 ESO UNIT 5
In this unit you will review how to:
• Calculate the perimeter and the area of a polygon• Calculate the perimeter and the area of a circular shape• Calculate the area of a prism• Calculate the area of a pyramid• Calculate the area of a polyhedron• Calculate the area of a solid of revolution• Calculate the volume of a 3-D shape
In this unit you will learn how to: • Identify similar shapes• Calculate the ratio of similarity of similar shapes• Calculate the ratio of proportionality between areas of similar shapes• Calculate the ratio of proportionality between volumes of similar shapes
P = n · l
the base by using the Pythagoras’ theorem:
22 2426 −
166
EXERCISES
52
tripas_MATHEMATICS_4.indd 166 23/08/2019 11:32:17
1. Polygons. Perimeter of a polygon
perimeter
“ :”
In the case of a regular polygonperimeter P
l
PERIMETERS, SURFACES AND VOLUMES. SIMILARITIES 4 ESO UNIT 5
In this unit you will review how to:
• Calculate the perimeter and the area of a polygon• Calculate the perimeter and the area of a circular shape• Calculate the area of a prism• Calculate the area of a pyramid• Calculate the area of a polyhedron• Calculate the area of a solid of revolution• Calculate the volume of a 3-D shape
In this unit you will learn how to: • Identify similar shapes• Calculate the ratio of similarity of similar shapes• Calculate the ratio of proportionality between areas of similar shapes• Calculate the ratio of proportionality between volumes of similar shapes
P = n · l
the base by using the Pythagoras’ theorem:
22 2426 −
167
EXERCISES
prisms
432
pyramids
tripas_MATHEMATICS_4.indd 167 23/08/2019 11:32:17
UNIT 5:
“PERIMETERS, SURFACES AND VOLUMES.
SIMILARITIES”
UNIT 5 PERIMETERS, SURFACES AND VOLUMES. SIMILARITIES 4 ESO
KEY VOCABULARY:
Polygon Regular Irregular Perimeter Area Surface Quadrilateral Composite Apothem Circular Sector Segment Annulus Polyhedron Platonic solid Prism Cuboid Convex Concave Pyramid Apex Surface development Tetrahedron Solid of revolution Axis of revolution Cylinder Cone Sphere Spherical cap Spherical wedge Ungula Similar polygons Ratio of similarity Ratio of proportionality
1. Polygons. Perimeter of a polygon
perimeter
“ :”
In the case of a regular polygonperimeter P
l
PERIMETERS, SURFACES AND VOLUMES. SIMILARITIES 4 ESO UNIT 5
In this unit you will review how to:
• Calculate the perimeter and the area of a polygon• Calculate the perimeter and the area of a circular shape• Calculate the area of a prism• Calculate the area of a pyramid• Calculate the area of a polyhedron• Calculate the area of a solid of revolution• Calculate the volume of a 3-D shape
In this unit you will learn how to: • Identify similar shapes• Calculate the ratio of similarity of similar shapes• Calculate the ratio of proportionality between areas of similar shapes• Calculate the ratio of proportionality between volumes of similar shapes
P = n · l
the base by using the Pythagoras’ theorem:
22 2426 −
168
EXERCISES
composite 3-D shapes
solids of revolution
tripas_MATHEMATICS_4.indd 168 23/08/2019 11:32:18
1. Polygons. Perimeter of a polygon
perimeter
“ :”
In the case of a regular polygonperimeter P
l
PERIMETERS, SURFACES AND VOLUMES. SIMILARITIES 4 ESO UNIT 5
In this unit you will review how to:
• Calculate the perimeter and the area of a polygon• Calculate the perimeter and the area of a circular shape• Calculate the area of a prism• Calculate the area of a pyramid• Calculate the area of a polyhedron• Calculate the area of a solid of revolution• Calculate the volume of a 3-D shape
In this unit you will learn how to: • Identify similar shapes• Calculate the ratio of similarity of similar shapes• Calculate the ratio of proportionality between areas of similar shapes• Calculate the ratio of proportionality between volumes of similar shapes
P = n · l
the base by using the Pythagoras’ theorem:
22 2426 −
169
EXERCISES
spherical shapes
15cm
5cm
tripas_MATHEMATICS_4.indd 169 23/08/2019 11:32:19
UNIT 5:
“PERIMETERS, SURFACES AND VOLUMES.
SIMILARITIES”
UNIT 5 PERIMETERS, SURFACES AND VOLUMES. SIMILARITIES 4 ESO
KEY VOCABULARY:
Polygon Regular Irregular Perimeter Area Surface Quadrilateral Composite Apothem Circular Sector Segment Annulus Polyhedron Platonic solid Prism Cuboid Convex Concave Pyramid Apex Surface development Tetrahedron Solid of revolution Axis of revolution Cylinder Cone Sphere Spherical cap Spherical wedge Ungula Similar polygons Ratio of similarity Ratio of proportionality
1. Polygons. Perimeter of a polygon
perimeter
“ :”
In the case of a regular polygonperimeter P
l
PERIMETERS, SURFACES AND VOLUMES. SIMILARITIES 4 ESO UNIT 5
In this unit you will review how to:
• Calculate the perimeter and the area of a polygon• Calculate the perimeter and the area of a circular shape• Calculate the area of a prism• Calculate the area of a pyramid• Calculate the area of a polyhedron• Calculate the area of a solid of revolution• Calculate the volume of a 3-D shape
In this unit you will learn how to: • Identify similar shapes• Calculate the ratio of similarity of similar shapes• Calculate the ratio of proportionality between areas of similar shapes• Calculate the ratio of proportionality between volumes of similar shapes
P = n · l
the base by using the Pythagoras’ theorem:
22 2426 −
170
EXERCISES
tripas_MATHEMATICS_4.indd 170 23/08/2019 11:32:20
1. Polygons. Perimeter of a polygon
perimeter
“ :”
In the case of a regular polygonperimeter P
l
PERIMETERS, SURFACES AND VOLUMES. SIMILARITIES 4 ESO UNIT 5
In this unit you will review how to:
• Calculate the perimeter and the area of a polygon• Calculate the perimeter and the area of a circular shape• Calculate the area of a prism• Calculate the area of a pyramid• Calculate the area of a polyhedron• Calculate the area of a solid of revolution• Calculate the volume of a 3-D shape
In this unit you will learn how to: • Identify similar shapes• Calculate the ratio of similarity of similar shapes• Calculate the ratio of proportionality between areas of similar shapes• Calculate the ratio of proportionality between volumes of similar shapes
P = n · l
the base by using the Pythagoras’ theorem:
22 2426 −
171
EXERCISES
Truncated shapes
Two tips
• Pythagoras’ theorem
• Thales’ theorem
2
2ba
−
c'c
b'b
a'a ==
tripas_MATHEMATICS_4.indd 171 23/08/2019 11:32:20
UNIT 5:
“PERIMETERS, SURFACES AND VOLUMES.
SIMILARITIES”
UNIT 5 PERIMETERS, SURFACES AND VOLUMES. SIMILARITIES 4 ESO
KEY VOCABULARY:
Polygon Regular Irregular Perimeter Area Surface Quadrilateral Composite Apothem Circular Sector Segment Annulus Polyhedron Platonic solid Prism Cuboid Convex Concave Pyramid Apex Surface development Tetrahedron Solid of revolution Axis of revolution Cylinder Cone Sphere Spherical cap Spherical wedge Ungula Similar polygons Ratio of similarity Ratio of proportionality
1. Polygons. Perimeter of a polygon
perimeter
“ :”
In the case of a regular polygonperimeter P
l
PERIMETERS, SURFACES AND VOLUMES. SIMILARITIES 4 ESO UNIT 5
In this unit you will review how to:
• Calculate the perimeter and the area of a polygon• Calculate the perimeter and the area of a circular shape• Calculate the area of a prism• Calculate the area of a pyramid• Calculate the area of a polyhedron• Calculate the area of a solid of revolution• Calculate the volume of a 3-D shape
In this unit you will learn how to: • Identify similar shapes• Calculate the ratio of similarity of similar shapes• Calculate the ratio of proportionality between areas of similar shapes• Calculate the ratio of proportionality between volumes of similar shapes
P = n · l
the base by using the Pythagoras’ theorem:
22 2426 −
172
EXERCISES
You can calculate A in this way:
A = A big cone - A small cone + upper base
ONE TIP:
3cm
tripas_MATHEMATICS_4.indd 172 23/08/2019 11:32:21
1. Polygons. Perimeter of a polygon
perimeter
“ :”
In the case of a regular polygonperimeter P
l
PERIMETERS, SURFACES AND VOLUMES. SIMILARITIES 4 ESO UNIT 5
In this unit you will review how to:
• Calculate the perimeter and the area of a polygon• Calculate the perimeter and the area of a circular shape• Calculate the area of a prism• Calculate the area of a pyramid• Calculate the area of a polyhedron• Calculate the area of a solid of revolution• Calculate the volume of a 3-D shape
In this unit you will learn how to: • Identify similar shapes• Calculate the ratio of similarity of similar shapes• Calculate the ratio of proportionality between areas of similar shapes• Calculate the ratio of proportionality between volumes of similar shapes
P = n · l
the base by using the Pythagoras’ theorem:
22 2426 −
173
EXERCISES
Truncated shapes
tripas_MATHEMATICS_4.indd 173 23/08/2019 11:32:22
UNIT 5:
“PERIMETERS, SURFACES AND VOLUMES.
SIMILARITIES”
UNIT 5 PERIMETERS, SURFACES AND VOLUMES. SIMILARITIES 4 ESO
KEY VOCABULARY:
Polygon Regular Irregular Perimeter Area Surface Quadrilateral Composite Apothem Circular Sector Segment Annulus Polyhedron Platonic solid Prism Cuboid Convex Concave Pyramid Apex Surface development Tetrahedron Solid of revolution Axis of revolution Cylinder Cone Sphere Spherical cap Spherical wedge Ungula Similar polygons Ratio of similarity Ratio of proportionality
1. Polygons. Perimeter of a polygon
perimeter
“ :”
In the case of a regular polygonperimeter P
l
PERIMETERS, SURFACES AND VOLUMES. SIMILARITIES 4 ESO UNIT 5
In this unit you will review how to:
• Calculate the perimeter and the area of a polygon• Calculate the perimeter and the area of a circular shape• Calculate the area of a prism• Calculate the area of a pyramid• Calculate the area of a polyhedron• Calculate the area of a solid of revolution• Calculate the volume of a 3-D shape
In this unit you will learn how to: • Identify similar shapes• Calculate the ratio of similarity of similar shapes• Calculate the ratio of proportionality between areas of similar shapes• Calculate the ratio of proportionality between volumes of similar shapes
P = n · l
the base by using the Pythagoras’ theorem:
22 2426 −
174
EXERCISES
if you know that the area of the biggest one is 36πcm
sphere is 100πcm
tripas_MATHEMATICS_4.indd 174 23/08/2019 11:32:22
1. Polygons. Perimeter of a polygon
perimeter
“ :”
In the case of a regular polygonperimeter P
l
PERIMETERS, SURFACES AND VOLUMES. SIMILARITIES 4 ESO UNIT 5
In this unit you will review how to:
• Calculate the perimeter and the area of a polygon• Calculate the perimeter and the area of a circular shape• Calculate the area of a prism• Calculate the area of a pyramid• Calculate the area of a polyhedron• Calculate the area of a solid of revolution• Calculate the volume of a 3-D shape
In this unit you will learn how to: • Identify similar shapes• Calculate the ratio of similarity of similar shapes• Calculate the ratio of proportionality between areas of similar shapes• Calculate the ratio of proportionality between volumes of similar shapes
P = n · l
the base by using the Pythagoras’ theorem:
22 2426 −
175
EXERCISES
reduction scale
tripas_MATHEMATICS_4.indd 175 23/08/2019 11:32:23
Top Related