Comparing Cylinder and Cone Volumes | Grade 8 Math | LevelKite

You already know how to find the volume of a cylinder and the volume of a cone. In this lesson you put these two solids, or three-dimensional shapes, side by side and ask: which one fills more space, and how do the two amounts compare?

A cylinder is a solid with two matching circular bases that are parallel to each other. A cone is a solid with one circular base and a single point, called the apex, opposite the base. Volume is the amount of space a solid fills, counted in cubic units such as cubic centimeters.

To compare two volumes written in the same cubic unit, you can subtract to find how many cubic units more one solid fills, or you can divide to find how many times as large one volume is as the other. You will do both.

Start with a pair you could hold: a can and a cone-shaped paper cup. The can is a cylinder and the cup is a cone. Suppose they have the same radius and the same height. Here the height is the perpendicular one, not the slant height of the cup.

Fill the cup with water and pour it into the can. The can is not full. A second cup does not fill it either. A third cup fills the can exactly to the top. For this matching pair, the cylinder holds three times as much as the cone, and the cone holds one third as much as the cylinder.

The two volume formulas you have already used show why the number is exactly three.

V_{\text{cylinder}} = \pi r^2 h \qquad\qquad V_{\text{cone}} = \tfrac{1}{3}\pi r^2 h

Both formulas contain $\pi r^2 h$ . The cone's formula has one extra factor, $\tfrac{1}{3}$ . A factor is a number you multiply by.

Now take a cylinder and a cone with the same radius $r$ and the same perpendicular height $h$ , in the same unit, like the can and the cup. Then $\pi r^2 h$ is the very same amount for both solids. It is the cylinder's whole volume, and the cone's volume is $\tfrac{1}{3}$ of it. Three cones together hold $3 \cdot \tfrac{1}{3}\pi r^2 h$ , and because $3 \cdot \tfrac{1}{3} = 1$ , that is exactly $\pi r^2 h$ , the cylinder's volume. That is why three cups filled the can.

The table checks this with numbers. In each row the cylinder and the cone have the same radius and the same height. In the last row the cone's volume is a fraction, yet three of those cones still fill the cylinder, because $3 \cdot \tfrac{20}{3}\pi = 20\pi$ .

Cylinder and cone pairs with the same radius and the same perpendicular height

Radius (cm)	Height (cm)	Cylinder volume (cubic cm)	Cone volume (cubic cm)	Cylinder volume divided by cone volume
$1$	$3$	$\pi \cdot 1^2 \cdot 3 = 3\pi$	$\tfrac{1}{3} \cdot 3\pi = \pi$	$3$
$2$	$6$	$\pi \cdot 2^2 \cdot 6 = 24\pi$	$\tfrac{1}{3} \cdot 24\pi = 8\pi$	$3$
$5$	$12$	$\pi \cdot 5^2 \cdot 12 = 300\pi$	$\tfrac{1}{3} \cdot 300\pi = 100\pi$	$3$
$2$	$5$	$\pi \cdot 2^2 \cdot 5 = 20\pi$	$\tfrac{1}{3} \cdot 20\pi = \tfrac{20}{3}\pi$	$3$

Key concept

Matching radius and height: the cone is one third

Suppose a cylinder and a cone have the same radius $r$ (so also the same diameter, because $d = 2r$ ) and the same height $h$ , measured perpendicular to the base and not along a slanted side. Then the cone's volume is exactly one third of the cylinder's volume, and the cylinder's volume is exactly three times the cone's volume. This holds for all positive $r$ and $h$ written in the same unit, because both volumes are built from the same $\pi r^2 h$ .

If the radius or the height is not the same for both solids, do not assume this relationship. Find each volume with its own formula, in the same cubic unit, and then compare the two results.

V_{\text{cone}} = \tfrac{1}{3} \cdot V_{\text{cylinder}} \quad \text{for the same } r \text{ and the same } h

Worked example

A matching cylinder and cone

A cylinder and a cone each have a radius of $3$ cm and a height of $10$ cm. Each height is measured perpendicular to the base. Compare their volumes.

Find the cylinder's volume with $V = \pi r^2 h$ . Square the radius first, then multiply by the height. Both lengths are in centimeters, so the volume is in cubic centimeters.
$V_{\text{cylinder}} = \pi \cdot 3^2 \cdot 10 = \pi \cdot 9 \cdot 10 = 90\pi \text{ cm}^3$
Find the cone's volume with $V = \tfrac{1}{3}\pi r^2 h$ . The cone has the same radius and the same height as the cylinder, so $\pi r^2 h$ is $90\pi$ again, and the cone's volume is one third of it.
$V_{\text{cone}} = \tfrac{1}{3} \cdot \pi \cdot 3^2 \cdot 10 = \tfrac{1}{3} \cdot 90\pi = 30\pi \text{ cm}^3$
Divide to find how many times as large the cylinder's volume is. Both volumes are in cubic centimeters, and $\pi$ is not zero, so the factor $\pi$ divides out of the top and the bottom.
$\dfrac{90\pi}{30\pi} = \dfrac{90}{30} = 3$
Subtract to find how much more space the cylinder fills. You can subtract because both volumes are in the same cubic unit.
$90\pi - 30\pi = 60\pi \text{ cm}^3$

Answer

The cylinder's volume, $90\pi$ cubic centimeters, is exactly $3$ times the cone's volume, $30\pi$ cubic centimeters. The cylinder fills $60\pi$ cubic centimeters more space than the cone.

Many pairs do not match. The cylinder and the cone may differ in radius, in height, or in both. Then you cannot rely on the one-third relationship, and you compare in two moves.

Find each volume with its own formula. Use each solid's radius and perpendicular height, with every length in the same unit.
Compare the two results. Subtract to find how many cubic units more one solid fills. Divide to find how many times as large one volume is as the other.

Keep $\pi$ as a symbol while you work. When both volumes are written as a number times $\pi$ in the same cubic unit, the one with the larger number is the larger volume, because $\pi$ is positive.

Do not use one third for every pair. A cylinder has radius $3$ cm and a cone has radius $6$ cm. Both have height $4$ cm. A common mistake is to find the cylinder's volume, $\pi \cdot 3^2 \cdot 4 = 36\pi$ cubic centimeters, then say the cone holds a third of that, $12\pi$ cubic centimeters.

The cone's own formula gives $\tfrac{1}{3} \cdot \pi \cdot 6^2 \cdot 4 = \tfrac{1}{3} \cdot 144\pi = 48\pi$ cubic centimeters. That is more than the cylinder holds.

This fails because the one-third relationship needs both solids to share the same $\pi r^2 h$ . Here the radius is $3$ cm for one solid and $6$ cm for the other, so $\pi r^2 h$ is $36\pi$ for the cylinder but $144\pi$ for the cone. Use one third only when the radius and the perpendicular height both match.

Worked example

Different measurements

A cylinder has a diameter of $8$ cm and a height of $9$ cm. A cone has a radius of $6$ cm, a height of $8$ cm and a slant height of $10$ cm. Each height is measured perpendicular to the base. Which solid has the greater volume, and how do the two volumes compare?

The cylinder's diameter is given, but the formula needs its radius. Because $d = 2r$ , divide the diameter by $2$ .
$r = d \div 2 = 8 \div 2 = 4 \text{ cm}$
Find the cylinder's volume with $V = \pi r^2 h$ , using $r = 4$ and $h = 9$ . Both lengths are in centimeters, so the volume is in cubic centimeters.
$V_{\text{cylinder}} = \pi \cdot 4^2 \cdot 9 = \pi \cdot 16 \cdot 9 = 144\pi \text{ cm}^3$
Find the cone's volume with $V = \tfrac{1}{3}\pi r^2 h$ , using $r = 6$ and $h = 8$ . The height of $8$ cm is perpendicular to the base. The slant height of $10$ cm runs along the sloping side, so the formula does not use it.
$V_{\text{cone}} = \tfrac{1}{3} \cdot \pi \cdot 6^2 \cdot 8 = \tfrac{1}{3} \cdot \pi \cdot 36 \cdot 8 = \tfrac{1}{3} \cdot 288\pi = 96\pi \text{ cm}^3$
Decide which volume is greater. Both are a number times $\pi$ in cubic centimeters, and $\pi$ is positive, so compare the numbers $144$ and $96$ .
$144 > 96 \quad \text{so} \quad 144\pi > 96\pi$
Subtract to find how much more space the cylinder fills. The difference $48\pi$ is exact because nothing has been rounded. For a decimal, replace $\pi$ with the rounded value $3.14$ . The decimal is an approximation, so say about $151$ cubic centimeters.
$144\pi - 96\pi = 48\pi \text{ cm}^3, \qquad 48\pi \approx 48 \times 3.14 = 150.72$
Divide to find how many times as large the cylinder's volume is. Because $\pi$ is not zero, it divides out. Then divide the top and the bottom by $48$ , which does not change the value of the fraction.
$\dfrac{144\pi}{96\pi} = \dfrac{144}{96} = \dfrac{3}{2} = 1.5$

Answer

The cylinder has the greater volume. It fills $48\pi$ cubic centimeters more than the cone, which is about $151$ cubic centimeters. Its volume is $1.5$ times the cone's volume, not $3$ times.

Worked example

A cone three times as tall

A cylinder and a cone have the same radius, $r$ centimeters, where $r$ is any positive number. The cylinder's height is $5$ cm. The cone's height is $15$ cm, which is $3$ times the cylinder's height. Each height is measured perpendicular to the base. How do the two volumes compare?

Write the cylinder's volume with $V = \pi r^2 h$ and $h = 5$ . Keep the radius as the letter $r$ , so the result is true for every positive radius. The volume is in cubic centimeters.
$V_{\text{cylinder}} = \pi \cdot r^2 \cdot 5 = 5\pi r^2 \text{ cm}^3$
Write the cone's volume with $V = \tfrac{1}{3}\pi r^2 h$ and $h = 15$ . Multiply the numbers first: one third of $15$ is $5$ .
$V_{\text{cone}} = \tfrac{1}{3} \cdot \pi \cdot r^2 \cdot 15 = \tfrac{15}{3} \cdot \pi r^2 = 5\pi r^2 \text{ cm}^3$
Compare the two results. They are the same expression, so the volumes are equal. Dividing shows it too: $r$ is positive, so $5\pi r^2$ is not zero and you may divide by it.
$\dfrac{5\pi r^2}{5\pi r^2} = 1$

Answer

The two volumes are equal: each is $5\pi r^2$ cubic centimeters for every positive radius $r$ . Making the cone $3$ times as tall exactly balances the factor $\tfrac{1}{3}$ in its formula.

Here is the idea to keep. When a cylinder and a cone have the same radius and the same perpendicular height, both positive lengths in the same unit, their volumes are built from the same $\pi r^2 h$ . So the cone's volume is exactly one third of the cylinder's, and the cylinder's is exactly three times the cone's.

When the radius or the height is not the same for both solids, do not assume that relationship. Find each volume from its own radius (half the diameter) and its own perpendicular height (never the slant height), keep $\pi$ for an exact answer, and then subtract to find how much more or divide to find how many times as large.