What shape can you

see?

SOLID GEOMETRY II

State the geometric properties of prisms, pyramids, cylinders, cones and spheres.

Draw nets for prisms, pyramids, cylinders and cones.

State and find surface areas of prisms, pyramids, cylinders, cones and spheres.

LEARNING OUTCOMES

Solid geometry is concerned with three-dimensional shapes.

Some examples of three-dimensional shapes are:

• Prisms• Pyramids• Cylinders• Cones• Spheres

DEFINITION

SOLIDS DESCRIPTION EXAMPLES

PRISM A solid with two congruent, parallel bases which are polygons.

PYRAMID A solid with a base which is a polygon and triangular sides that converge at a vertex.

CYLINDER A solid with two parallel congruent circular faces and a curved surface.

CONE A solid with a circular base and a vertex.

SPHERE A solid having all of its points the same distance from its centre.

12.1 PROPERTIES

Rectangular Prisms

Triangular Prisms

Hexagonal Prisms

Square Pyramids Rectangular Pyramid

Triangular Pyramid Hexagonal Pyramid

5 faces   8 edges  5 vertices  

2 faces   2 edges  1 vertices

5 faces   9 edges  6 vertices

12.2 NETS OF GEOMETRIC SOLIDS

12.2 NETS OF GEOMETRIC SOLIDS

A net is a two-dimensional figure that can be folded into a three-dimensional solid.

EXAMPLE 1

1)

2)

3)

4)

WORKSHEET

• It is measured using squares

• Units include mm²,cm²,m²,km².

The surface area of a solid is the total area of all the faces of the solid.

12.3 SURFACE AREA

SOLIDS NETS SURFACE AREA

PYRAMID

Area of four triangular faces + Area of rectangular base

PRISM

Area of three rectangular faces + Area of two triangular faces

Example 1:

Calculate the surface area of the pyramid shown.

SOLUTION

Area of square base

10 cm

13 cm21001010 cm

Area of a triangular face

26012102

1cm

Surface area of the pyramid

2340)604(100 cm

SURFACE AREA OF CYLINDER

r r

l h

l= circumference of the base circle r2

Area of curved surface (rectangular) + Area of two circular faces.

222 rrh

Example

Find the surface area of a cylinder with a radius of 7 cm and a height of 20 cm. (Take )

7

22

SOLUTION

cmr 7 cmh 20

Surface area of the cylinder

)20)(7)(7

22(2)7)(

7

22(2 2

21188880308 cm

rhr 22 2

SURFACE AREA OF CONE

l

r r

Area of sector =

Area of circle =2r

Area of sector + Area of circle

l

rl

2rrl

Example

Calculate the surface area of a cone with a radius of 5 cm and a slant height of 8 cm. (Take )142.3

SOLUTION

cmr 5

Surface area of the cone

)5)(142.3()8)(5)(142.3( 2

223.204 cm

cml 8

2rrl

SURFACE AREA OF SPHERE

Surface area of a sphere =24 r

Where r is the radius of the sphere

Example:

Find the surface area of the sphere. (Take )

7

22

SOLUTION

Surface area of the sphere:

222 1545.37

2244 cmr

POP QUIZ

1) Find the surface area of the sphere that has

a) radius =

b) diameter =

m11

31

cm8.2

SOLUTION

a)

b) Diameter =

11

31r

22

11

14

7

2244

r 3636.20

82.2

22 )2

8.2(

7

2244 r 64.24

2) Find the value of for the solid shown in the diagram if its surface area is 1551 .

Take

h

2cm

7

22

21 cm

h cm

SOLUTION

The solid given is cylinder.2

21r ?h

155122 2 rhr

15512

21

7

222

2

21

7

222

2

h

155166693 h

85866 h13h

3) A cone has a base of diameter 14 cm. Find the slant height of the cone if its surface area 286 .

Take

2cm

7

22

SOLUTION

Diameter =14 cm 7r ?l

2862 rrl

28677

227

7

22 2

s

28615422 s

13222 l

6l

4) A sphere has a surface area of .

What is its radius?

2

7

4804 mm

Let r be the radius of the sphere.

Surface area of the sphere = 24 r

22

7

48044 mmr

7

5632

7

224 2 r

7

5632

7

88 2 r

642 r

8r

5) Calculate the value of for the following solid.

x

10 cm

x cm

Surface area = 785 cm2

SOLUTION

10r

785107

2210

7

22 22

lrrl

7857

2200

7

220l

7

3295

7

220l

97.14l

6)

12 cm

5 cm

Calculate the surface area of the cone

Solution

13 cm12 cm

5 cm

Surface area = 282.8571

7) 2.8 mm

If the diameter of the iron rod is 2.8 mm and the surface area of the rod is 2.8mm, find its length.

Solution4.1r

32.8924.17

2224.1

7

22222 22

hrrh

32.89832.128.8 h

100h

Example 1Example 1

Find the total surface area of the following solid. Take . 3.142

The solid shown below consists of a cone and a hemisphere with a common base. What is the total surface area of the solid? Take . π 3.142

“Hemi” means half.

Example 2Example 2

• Ex12.3A, Ex12.3B, Ex12.3C

HOMEWORKHOMEWORK

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