8.03 Volume of cylinders
We have already seen how the volume of rectangular prisms can be calculated using the rule
$\text{Volume }=\text{Area of Base }\times\text{Height of Prism }$Volume =Area of Base ×Height of Prism
A cylinder is very similar to a prism (except that the base is a circle instead of a rectangle or other polygon), but the volume can be found using the same process. Check out this investigation for more detail on the derivation of the formula for the volume of a cylinder.
| $\text{Volume of Cylinder }$Volume of Cylinder | $=$= | $\text{Area of Base }\times\text{Height of Prism }$Area of Base ×Height of Prism |
| $\text{Volume of Cylinder }$Volume of Cylinder | $=$= | $\pi r^2\times h$πr2×h |
| $\text{Volume of Cylinder }$Volume of Cylinder | $=$= | $\pi r^2h$πr2h |
Worked example
Question 1
Find the volume of a cylinder with a diameter of $10$10 cm and a height of $7$7 cm. Round to two decimal places.
Think: The formula is $V=\pi r^2h$V=πr2h, we have been given $d=10$d=10 and $h=7$h=7, so we need to first find the radius and then we can substitute into the formula.
Do: The radius is half of the diameter, so $r=\frac{10}{2}$r=102 or $r=5$r=5.
| $V$V | $=$= | $\pi r^2h$πr2h |
Stating the formula |
| $=$= | $\pi\times5^2\times7$π×52×7 |
Substituting the given information |
|
| $=$= | $\pi\times25\times7$π×25×7 |
Evaluating the square first |
|
| $=$= | $549.78$549.78 |
Evaluating on a calculator and rounding |
The volume of this cylinder is $549.78$549.78 cm3.
Exploration
You are at the local hardware store to buy a can of paint. After settling on one product, the salesman offers to sell you a can of the same paint, but it will have either double the height or double the radius (your choice) for twice the price. Assuming all cans of paint are filled to the brim, is it worth taking him up on his offer?
If so, would you get more paint for each dollar if you chose the can that was double the radius or the can that was double the height?
To see how changes in height and radius affect the volume of a can to different extents, try the following interactive. You can vary the height and radius by moving the sliders around.
Practice questions
QUESTION 2
The volume of a cylinder is given by $V=\pi r^2h$V=πr2h.
Find the volume of the cylinder shown, rounding your answer to two decimal places.
QUESTION 3
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.
QUESTION 4
Consider the halfpipe with a diameter of $8$8 cm and a height of $20$20 cm.
Find its volume, rounding to two decimal places.