Volume of a pyramid

Volume of a pyramid =

1

3 base area perpendicular height

h

Calculate the volume of the rectangular-based pyramid.

6 cm

5 cm4

cm

Volume =

1

3 base area height

1

3(5 4) 6

40 cm3

A B

CD

E

Surface area of a pyramid

Surface area = sum of the areas of all the faces of the pyramidSurface area = sum of the areas of all the faces of the pyramid

h

Calculate the surface area of the rectangular-based pyramid.

6 cm

5 cm

4 cm

A B

CD

E First find the length of EX and EY.

EX2 2.52 62

EX2 42.25

EX 6.5

EY2 22 62

EY2 40

EY 6.325

Use Pythagoras on triangle EOX.

Use Pythagoras on triangle EOY.O X

Y

6.56.5

6.3256.325

5 cm 4 cm

A B

CD

E

E

E E

Surface area = sum of areas of faces

Area of rectangle ABCD = 4 × 5 = 20 cm2

Area of triangle BCE = ½ × 4 × 6.5 = 13 cm2

Area of triangle CDE = ½ × 5 × 6.325 = 15.81 cm2

= 20 + 13 + 13 + 15.81 + 15.81= 77.6 cm2

NET OFPYRAMID

Volume of a cone

Volume of a cone =

1

3 base area perpendicular height

1

3r 2h

h

r

Calculate the volume of the cone.

Volume =

1

3 base area height

1

3( 42) 7

117 cm3

7 cm

4 cm

Surface area of a cone

l h

r

+

The surface of a cone is made from a flat circular base and a curved surface.The curved surface is made from a sector of a circle.

FLAT BASE

CURVED SURFACE

= l l

Curved surface area of a cone = , where is the slant heightCurved surface area of a cone = , where is the slant height rl l

Total surface area of a cone = Total surface area of a cone = r 2 rl

Calculate a the curved surface area of the cone,b the total surface area of the cone.

12 c

m

5 cm

a First calculate the slant height using Pythagoras. l

l

l2

l2

lCurved surface area rl

5 13

204 cm2

b Total surface area r 2 rl

52 65

25 65

65

90 283 cm2

52 122

169

13

The straight edges of the sector are joined together to make a cone.Calculate a the curved surface area of the cone,

b the radius of the base of the cone,c the height of the cone. 280o

4 cm

4 cma Curved surface area = area of sector

280

360 42

39.1 cm2

112

9

b Curved surface area rl

39.1 r 4

r

39.1

4 r 3.11 cm

c Using Pythagoras

3.11

4 h h

2 3.112 42

h2 6.321

h 2.51 cm

When you make a cut parallel to the base of a cone and remove the top part, the part that is left is called a frustum.

FRUSTUM

Volume of frustum = volume of large cone – volume of smaller coneVolume of frustum = volume of large cone – volume of smaller cone

Calculate the volume of the frustum. All lengths are in cm.

3

6

8

h

You must first find the height of the smaller cone using similar triangles.

h

3

h 8

6

6h 3h 24

3h 24

h 8

Volume of large cone

1

3r 2h

1

3 62 16

192

Volume of small cone

1

3r 2h

1

3 32 8

24Volume of frustum 192 24

528 cm3

3

6

8

Volume and surface area of a sphere

Volume of a sphere Volume of a sphere

4

3r 3

Surface area of a sphereSurface area of a sphere 4r 2

Volume and surface area of a hemisphere

Volume of a hemisphere Volume of a hemisphere

2

3r 3

Curved surface area of a hemisphereCurved surface area of a hemisphere 2r 2

A hemisphere is half a sphere.

The sphere has radius 10 cm.Calculate a the volume of the sphere,

b the surface area of the sphere.

a Volume

4

3r 3

4

3 103

4189 cm3

b Surface area 4r 2

4 102

1257 cm2

The solid hemisphere has radius 6 cm.Calculatea the volume of the hemisphere,b the curved surface area of the hemisphere,c the total surface area of the hemisphere.

6 cm

a Volume

2

3r 3

2

3 63

452 cm3

b Curved surface area 2r 2

2 62

226 cm2

c Total surface area = area of base circle + curved surface area

62 226

339 cm2

The solid is made from a cylinder and a hemisphere.The cylinder has a height of 8 cm and a radius of 3 cm.Calculate the volume of the solid.

Volume of cylinder r 2h

32 8

72

Volume of hemisphere

2

3r 3

2

3 33

18

Total volume 72 18

90 283 cm3