Volume of Cylinders and Cones | Grade 8 Math | LevelKite

You already know that the volume of a prism is the area of its base times its height: $V = Bh$ , where $B$ is the base area and $h$ is the height measured at a right angle to the base. Volume is the amount of space a solid fills, and you measure it in cubic units such as cubic centimeters ( $\text{cm}^3$ ).

In this lesson you will stretch that one idea to two round solids. First you will find the volume of a cylinder. Then you will use the cylinder to find the volume of a cone.

A cylinder is a solid with two identical circular bases that are parallel to each other, joined by a curved surface. A soup can is shaped like a cylinder. Its volume depends on two lengths:

The radius $r$ is the distance from the center of a base to its edge. The diameter $d$ is the distance straight across a base through its center, so $d = 2r$ and $r = d \div 2$ .
The height $h$ is the distance between the two bases, measured perpendicular to them. Perpendicular means at a right angle.

In every problem, $r$ , $d$ and $h$ are positive lengths, and $r$ and $h$ must be in the same unit of length before you calculate. Lengths use linear units such as cm, areas use square units such as $\text{cm}^2$ , and volumes use cubic units such as $\text{cm}^3$ .

Why does the prism rule carry over to a cylinder? A prism has the same shape at every level, from its bottom base to its top. So does a cylinder: it is the same circle at every level. That means you can think of a cylinder in layers. A layer one unit thick on a base of area $B$ square units holds $B$ cubic units, and a height of $h$ units is worth $h$ of those layers. The volume is $B \times h$ , just as it is for a prism.

The base of a cylinder is a circle, and the area of a circle with radius $r$ is $\pi r^2$ . Replace $B$ with $\pi r^2$ to get the rule for a cylinder. It holds when $r$ is the radius of the base, $r$ and $h$ are positive lengths in the same unit, and $h$ is the perpendicular height:

V = B \cdot h = \pi r^2 \cdot h = \pi r^2 h \qquad (r > 0,\ h > 0)

Worked example

Volume of a cylinder

A cylinder has radius $3$ cm. Its height, perpendicular to the bases, is $5$ cm. Find the exact volume, then approximate it using $3.14$ for $\pi$ .

List the measurements and check the conditions. Both lengths are positive, both are in centimeters, and the height is the perpendicular height, so the cylinder rule applies.
$r = 3 \text{ cm}, \quad h = 5 \text{ cm}$
Find the area of the circular base. Square the radius first, because the exponent belongs to r only, not to π: 3 squared is 3 × 3 = 9. Centimeters times centimeters gives square centimeters.
$B = \pi r^2 = \pi \cdot 3^2 = 9\pi \text{ cm}^2$
Multiply the base area by the height, because a cylinder's volume is its base area times its height. The order of multiplication does not change a product, so multiply the numbers and keep π as a factor: 9 × 5 = 45. Square centimeters times centimeters gives cubic centimeters.
$V = B \cdot h = 9\pi \cdot 5 = 45\pi \text{ cm}^3$
Approximate by replacing π with 3.14. The result is no longer exact, so switch from = to ≈. Multiply in parts: 45 × 3 = 135 and 45 × 0.14 = 6.3, and 135 + 6.3 = 141.3.
$V \approx 45 \times 3.14 = 141.3 \text{ cm}^3$

Answer

The volume of the cylinder is exactly $45\pi \text{ cm}^3$ , which is about $141.3 \text{ cm}^3$ when you use $3.14$ for $\pi$ .

A cone is a solid with one circular base and a curved surface that narrows to a single point called the apex. In this lesson the apex is directly above the center of the base. The radius $r$ and the diameter $d$ of the base mean the same as before, with $d = 2r$ . A cone has two other lengths that are easy to mix up:

The height $h$ is the distance from the apex straight down to the center of the base. It is perpendicular to the base.
The slant height is the distance from the apex down the sloping side to the edge of the base. It is longer than $h$ , and it is not part of the volume rule.

How does a cone compare with a cylinder that has the same base and the same height? The cylinder is the same circle at every level. The cone begins with that circle at its base, but its circle shrinks at each level until it is a single point at the apex. So the cone must hold less than the cylinder.

The exact fraction is one third. If the two solids were hollow containers, it would take exactly three full cones of water to fill the cylinder. You can test this by pouring, and later mathematics proves it for every cone and cylinder that have the same base radius and the same perpendicular height. The table shows the pattern with numbers.

A cylinder and a cone with the same radius and the same perpendicular height

Radius	Height	Cylinder volume	Cone volume
$3$ cm	$5$ cm	$\pi \cdot 3^2 \cdot 5 = 45\pi \text{ cm}^3$	$45\pi \div 3 = 15\pi \text{ cm}^3$
$2$ cm	$9$ cm	$\pi \cdot 2^2 \cdot 9 = 36\pi \text{ cm}^3$	$36\pi \div 3 = 12\pi \text{ cm}^3$
$6$ cm	$4$ cm	$\pi \cdot 6^2 \cdot 4 = 144\pi \text{ cm}^3$	$144\pi \div 3 = 48\pi \text{ cm}^3$

Key concept

Base area times height, and one third of it

The volume of a cylinder is the area of its circular base times its height. A cone with the same base radius and the same height has exactly one third of that volume.

These rules hold when:

$r$ is the radius of the base, so halve a diameter first: $r = d \div 2$ ;
$h$ is the height measured perpendicular to the base, never the slant height;
$r$ and $h$ are positive lengths in the same unit.

The volume is in cubic units. Keep $\pi$ for an exact answer, or use $3.14$ for an approximation and write $\approx$ .

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

Worked example

Volume of a cone

A cone has radius $4$ m. Its height, perpendicular to the base, is $6$ m. Find the exact volume, then approximate it using $3.14$ for $\pi$ .

List the measurements and check the conditions. Both lengths are positive, both are in meters, and the height is the perpendicular height, so the cone rule applies.
$r = 4 \text{ m}, \quad h = 6 \text{ m}$
Start with the cylinder that has the same base and the same height: base area times height. 4 squared is 4 × 4 = 16, and 16 × 6 = 96. Square meters times meters gives cubic meters.
$\pi r^2 h = \pi \cdot 4^2 \cdot 6 = \pi \cdot 16 \cdot 6 = 96\pi \text{ m}^3$
Take one third of that, because the cone holds one third of the matching cylinder. Multiplying by one third is the same as dividing by 3, and 96 ÷ 3 = 32.
$V = \tfrac{1}{3} \cdot 96\pi = 32\pi \text{ m}^3$
Approximate by replacing π with 3.14 and switching to ≈, because the result is no longer exact. Multiply in parts: 32 × 3 = 96 and 32 × 0.14 = 4.48, and 96 + 4.48 = 100.48.
$V \approx 32 \times 3.14 = 100.48 \text{ m}^3$

Answer

The volume of the cone is exactly $32\pi \text{ m}^3$ , which is about $100.48 \text{ m}^3$ when you use $3.14$ for $\pi$ .

Worked example

A cone with a diameter and a slant height

A cone has a base with diameter $12$ cm. Its height, from the apex straight down to the center of the base, is $8$ cm. Its slant height is $10$ cm. Find the exact volume, then approximate it using $3.14$ for $\pi$ .

Choose the height the rule needs. The cone rule uses the perpendicular height, which is 8 cm. The slant height of 10 cm runs along the sloping side, so set it aside.
$h = 8 \text{ cm}$
Turn the diameter into a radius, because the rule uses the radius and a diameter is two radii. Now both lengths are positive and in centimeters, so the cone rule applies.
$r = d \div 2 = 12 \div 2 = 6 \text{ cm}$
Substitute the radius and the height into the cone rule, then square the radius first: 6 squared is 6 × 6 = 36.
$V = \tfrac{1}{3}\pi r^2 h = \tfrac{1}{3} \cdot \pi \cdot 6^2 \cdot 8 = \tfrac{1}{3} \cdot \pi \cdot 36 \cdot 8$
Multiply the numbers and keep π as a factor: 36 × 8 = 288. Then take one third, which is the same as dividing by 3: 288 ÷ 3 = 96. Square centimeters times centimeters gives cubic centimeters.
$V = \tfrac{1}{3} \cdot 288\pi = 96\pi \text{ cm}^3$
Approximate by replacing π with 3.14 and switching to ≈: 96 × 3.14 = 301.44.
$V \approx 96 \times 3.14 = 301.44 \text{ cm}^3$

Answer

The volume of the cone is exactly $96\pi \text{ cm}^3$ , which is about $301.44 \text{ cm}^3$ when you use $3.14$ for $\pi$ .

Takeaway: A cylinder's volume is its base area times its height, $V = \pi r^2 h$ . A cone with the same base radius and the same height holds exactly one third as much, $V = \tfrac{1}{3}\pi r^2 h$ . Both rules need a positive radius and a positive perpendicular height in the same unit of length, and both give cubic units. Before you calculate, halve any diameter and set aside any slant height.

Next, you will run these rules backward to find a missing dimension when you know the volume, solve word problems, and find and correct errors in volume calculations.