In this section we look at the volumes of cylinders and cones.
Cylinders
We have already seen how the volume of prisms can be calculated using the rule:
$\text{Volume }=\text{area of base }\times\text{height }$Volume =area of base ×height
The base of a prism is the face that is extended, or extruded, throughout the height. Another way of identifying the base is finding the two faces that are identical and parallel to one another.
A cylinder is very similar to a prism (except for the circular face), but the volume can be found using the same process. We can think of the circle as the face that is extended throughout its height!
| Volume of a cylinder | $=$= | $\text{area of base }\times\text{height }$area of base ×height |
| $V$V | $=$= | $\pi r^2\times h$πr2×h |
| $V$V | $=$= | $\pi r^2h$πr2h |
Practice question
Question 1
Find the volume of a cylinder correct to one decimal place if its radius is $6$6 cm and its height is $15$15 cm.
Volumes formed from part of a cylinder
When we are asked to find the volume of half or quarter of a cylinder we simply multiply the full volume by the fraction we are trying to find.
Worked example
A solid is formed by extruding a sector, as shown in the diagram below. The sector has a radius of $4$4 m and is extruded to a height of $10$10 m. Find the volume of the solid, to the nearest cm2.
Think: The shaded sector has an angle at the centre of $78^\circ$78°. When the sector is extruded it forms a fraction of cylinder given by $\frac{78}{360}$78360.
Do: we can do our calculation using this fraction of the whole volume, that is
| Volume of solid | $=$= | $\frac{78}{360}\times\text{volume of cylinder }$78360×volume of cylinder |
| $V$V | $=$= | $\frac{78}{360}\times\pi r^2h$78360×πr2h |
| $=$= | $\frac{78}{360}\times\pi\times4^2\times10$78360×π×42×10 | |
| $=$= | $108.9085$108.9085... | |
| $=$= | $109$109 cm2 (nearest cm2) |
Practice questions
QUESTION 2
Calculate the volume of the solid. Assume that the solid is a quarter of a cylinder.
Round your answer to one decimal place.
