1. Mod 47: Surface Area and Volume
Overview Find the surface area and volume of cones, pyramids and spheres Consider formulae for length, area and volume using dimensions.

2. height, h
length, l
width, w

3. A l h r

4. 1) 2) 3) 4) 6) 5)

5. Pyramids The most common pyramids are:
A tetrahedron or triangular pyramid.
A square-based pyramid
A cone
Ask pupils to tell you the shape of the faces rising up from the base when the base is a polygon.

6. Volume of a pyramid slant height h A base
Stress that the height must be perpendicular from the base to the apex.
Problems often give the slant height of a pyramid. The perpendicular height must then be found using Pythagoras’ Theorem.
Volume of a pyramid = × area of base × height 1 3
V = Ah 1 3

7. What is the volume of this rectangle-based pyramid?
Volume of a pyramid
What is the volume of this rectangle-based pyramid?
8 cm
5 cm
3 cm
Area of the base = 5 × 3
= 15 cm2
Volume of pyramid = Ah 1 3 1 3
= × 15 × 8
= 40 cm3

8. 2) 1) 3) 4) 7 cm 14 cm 9 cm 5 cm 6 cm 8 cm 8 cm 30 cm 3.3 cm 15 cm

9. Volume of a cone
A cone is a special type of pyramid with a circular base.
Remember, the volume of a pyramid can be found by multiplying the area of the base by the height and dividing by 3.
The volume of a cone is given by: r h
Volume = × area of circular base × height 1 3 or
V = πr2h 1 3

10. 2) 1) 3 8 5 2 3) 4) 7 12 4 3

11. Sphere

12. Volume and surface area of a sphere
For a sphere of radius r:
Volume = πr3 4 3 r
A sphere can be described as the 3-D locus of the points that are a fixed distance from a point.
These formulae will usually be given in an examination and it is not necessary for pupils to learn or derive them.
and
Surface area = 4πr2

13. Find the volumes and surface areas of the following solids:b) c) a)
r = 5cm
r = 7cm
r = 2.5m d) e)
d = 8 cm
d = 20 m

14. Dimensions

15. Dimensions of length, area and volume
one dimension
length
UNITS:
mm, cm, m, km,
area
two dimensions
length × length
UNITS:
mm2, cm2, m2, km2,
Use this table to discuss the differences between the three dimensions of length. Be careful to explain that if lengths are added rather than multiplied, the answer will still be one-dimensional. For example, perimeter has one dimension even though it is a measurement used for a two-dimensional shape. Similarly, surface area involves adding areas and has two dimensions even though it is a measurement used for a three-dimensional shape.
volume
three dimensions
length × length × length
UNITS:
mm3, cm3, m3, km3