1. SOLIDS & VOLUME
Common 3D Shapes
sphere
cuboid
cylinder
cube
cone

2. SPHERES
diameter
radius
A sphere is like a circle in 3D so can have a radius and diameter.
So calculations involving spheres use .
It can be shown that the volume occupied/filled by a sphere is given by the formula
V = 4/3r3

3. Example
12cm
V = 4/3r3
= 4/3 123
= 4  3 x  x 12 ^ 3
r = 12cm
= cm3

4. Example
20m
V = 4/3r3
= 4/3103
= 4  3 x  x 10 ^ 3
= m3
d = 20m so r = 10m

5. HEMISPHERES
A hemisphere is a half sphere. The equator divides the earth into the northern & southern hemispheres.
For a sphere we have V = 4/3r3
So for a hemisphere V = 2/3r3

6. Example
An igloo has a diameter of 3m.
Find its volume.
d = 3m so r = 1.5m
V = 2/3r3
= 2  3 x  x 1.5 ^ 3
= 7.069m3

7. CONES
radius
It can be shown that the volume of a cone is given by
height
V =1/3r2h
This is the same as
Vol = 1/3 X area base X height

8. Example
2cm
A cone for ice-cream is 10cm deep and has a radius of 2cm.
If ice-cream is packed inside this how much will it hold?
10cm
V =1/3r2h
= 3.14 X 2 X 2 X 10  3
= …
= 41.9cm3
( to 3 sfs )

9. Example
A traffic cone is 40cm high with a base of diameter 25cm.
40cm
Find its volume!
NB: d= 25 so r =12.5
25cm
V =1/3r2h
= 3.14 X 12.5 X 12.5 X 40  3
= …
= 6540cm3
( to 3 sfs )

10. Example
A pencil sharpener consists of a metal cuboid with a conical hole drilled out.
The sharpener is 2cm long, 1.5cm wide and 1cm thick
The hole is 2cm long with a diameter of 0.8cm.
Find the volume of metal in the sharpener.
*************
Cuboid
Cone
Vol metal
V = l X b X h
d = 0.8 so r = 0.4
= 3 – 0.334…
= 2 X 1.5 X 1
V =1/3r2h
= 2.665…
= 3.14 X 0.4 X 0.4 X 2  3
= 3cm3
= 2.67cm3
= ….cm3