How is the volume of a cone different from the volume of a cylinder?

A cone and a cylinder with the same radius and height have different volumes because the cone tapers to a point. Specifically, the cone's volume is exactly one-third of the cylinder's volume—hence the \frac{1}{3} in the formula.

What do I do if I'm only given the slant height?

The slant height is the distance along the cone's surface from the base edge to the tip, not the perpendicular height used in the formula. Use the Pythagorean theorem with h^2 + r^2 = l^2 (where l is slant height) to find the perpendicular height h.

Can the volume of a cone be negative?

No, volume is always positive. Since r, h, and \pi are all positive, V = \frac{1}{3}\pi r^2 h must be positive. If you get a negative answer, check that you used positive values for radius and height.

How do I find the radius if I only know the volume and height?

Rearrange the formula to solve for r: V = \frac{1}{3}\pi r^2 h becomes r^2 = \frac{3V}{\pi h}, then r = \sqrt{\frac{3V}{\pi h}}. Make sure to take the positive square root since radius is always positive.

Volume of a Cone

Calculate the space inside a cone by multiplying one-third of the base area by the height; essential for geometry and SAT problems.

V = \frac{1}{3}\pi r^2 h

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What each symbol means

What Volume of a Cone takes

Symbol	Meaning
$V$	The volume of the cone, measured in cubic units (cubic centimeters, cubic inches, cubic meters, etc.); this is what you're solving for.
$r$	The radius of the cone's circular base, measured in the same units as the height; it is always positive and half the diameter.
$h$	The height of the cone, measured perpendicular from the base to the apex (tip), in the same units as the radius; it is always positive.

When to use it

Use this when you need to find how much space a cone-shaped container can hold.

Level

Usually taught in: Geometry · Appears on: SAT

Worked examples

1. Find the volume of a cone with radius 3 cm and height 4 cm

Problem

A cone has a radius of 3 cm and a height of 4 cm. Find its volume.

$V = \frac{1}{3}\pi r^2 h$
Write the formula for volume of a cone.
$V = \frac{1}{3}\pi (3)^2 (4)$
Substitute $r = 3$ cm and $h = 4$ cm into the formula.
$V = \frac{1}{3}\pi \cdot 9 \cdot 4$
Calculate $3^2 = 9$ .
$V = \frac{1}{3}\pi \cdot 36$
Multiply the numbers: $9 \times 4 = 36$ .
$V = 12\pi \text{ cm}^3$
Divide $36$ by $3$ to get $\frac{36}{3} = 12$ .

Answer: $V = 12\pi \text{ cm}^3$

We substituted the given radius and height directly into the cone volume formula, then simplified by following the order of operations: squaring the radius, multiplying, and dividing by 3.

2. Find the volume when given diameter instead of radius

Problem

A cone has a diameter of 10 cm and a height of 6 cm. Find its volume.

$r = \frac{d}{2} = \frac{10}{2} = 5$
Since we're given the diameter, convert it to radius by dividing by 2: $d = 10$ cm, so $r = 5$ cm.
$V = \frac{1}{3}\pi r^2 h$
Write the volume formula.
$V = \frac{1}{3}\pi (5)^2 (6)$
Substitute $r = 5$ cm and $h = 6$ cm.
$V = \frac{1}{3}\pi \cdot 25 \cdot 6$
Calculate $5^2 = 25$ .
$V = \frac{1}{3}\pi \cdot 150$
Multiply $25 \times 6 = 150$ .
$V = 50\pi \text{ cm}^3$
Divide $150$ by $3$ to get $\frac{150}{3} = 50$ .

Answer: $V = 50\pi \text{ cm}^3$

When given diameter instead of radius, the first step is to divide by 2. This is a common setup for cone volume problems, and forgetting this conversion step is a frequent mistake.

3. Compare cone volumes for concert souvenir cups with two ticket tiers

Problem

A concert venue sells two types of cone-shaped souvenir cups. Standard ticket holders get cups with a radius of 3 inches and height of 7 inches. VIP ticket holders get cups with the same radius but a height of 10 inches. What is the volume difference between the two cup sizes?

$V_{\text{standard}} = \frac{1}{3}\pi r^2 h$
Write the formula using $r = 3$ and $h = 7$ .
$V_{\text{standard}} = \frac{1}{3}\pi (3)^2 (7)$
Substitute the standard cup dimensions.
$V_{\text{standard}} = \frac{1}{3}\pi \cdot 9 \cdot 7$
Calculate $3^2 = 9$ .
$V_{\text{standard}} = \frac{1}{3}\pi \cdot 63$
Multiply $9 \times 7 = 63$ .
$V_{\text{standard}} = 21\pi \text{ in}^3$
Divide $63$ by $3$ to get $\frac{63}{3} = 21$ .
$V_{\text{VIP}} = \frac{1}{3}\pi (3)^2 (10)$
Calculate the VIP cup volume with $r = 3$ and $h = 10$ .
$V_{\text{VIP}} = \frac{1}{3}\pi \cdot 9 \cdot 10$
Calculate $3^2 = 9$ .
$V_{\text{VIP}} = \frac{1}{3}\pi \cdot 90$
Multiply $9 \times 10 = 90$ .
$V_{\text{VIP}} = 30\pi \text{ in}^3$
Divide $90$ by $3$ to get $\frac{90}{3} = 30$ .
$\Delta V = 30\pi - 21\pi = 9\pi \text{ in}^3$
Find the difference between the two volumes: $30\pi - 21\pi = 9\pi$ .

Answer: $\Delta V = 9\pi \text{ in}^3$

This problem combines cone volume with a real-world scenario. By calculating both volumes separately, we can compare how changing only the height (while keeping radius constant) affects the total volume; in this case, a 3-inch height increase creates a $9\pi$ cubic inch capacity difference.

Common mistakes

Where Volume of a Cone usually goes wrong

The mistake	Why it is wrong	The fix
Using the full diameter instead of the radius in the formula.	The formula specifically requires the radius, not the diameter; using diameter gives a volume four times too large.	Always divide the diameter by 2 to get the radius before substituting into $V = \frac{1}{3}\pi r^2 h$ .
Forgetting the $\frac{1}{3}$ factor in the formula.	A cone's volume is exactly one-third of a cylinder with the same base and height; leaving out the $\frac{1}{3}$ makes your answer three times too large.	Remember that $V = \frac{1}{3}\pi r^2 h$ , not $V = \pi r^2 h$ (that would be a cylinder).
Confusing the slant height with the perpendicular height.	The perpendicular height (from base to apex) is what goes in the formula; slant height is the distance along the cone's side and is longer than the perpendicular height.	Always use the height measured perpendicular to the base, not the slant height; if only slant height is given, use the Pythagorean theorem with the radius to find it.

Tips and when to use something else

A cone has one-third the volume of a cylinder with the same base and height—remember the $\frac{1}{3}$ factor.
If you're given the diameter, always divide by 2 to get the radius before using the formula.
When you only know the slant height, use the Pythagorean theorem: $h^2 + r^2 = l^2$ , where $l$ is slant height.
For a cone inside or around other shapes, consider using Volume of a Cylinder or Volume of a Pyramid as comparison formulas.

Related formulas

Volume of a CylinderCalculates the volume of a cylinder—the three-dimensional space inside—used for any cylindrical container or object in math and engineering.Volume of a PyramidFind the volume inside a pyramid by multiplying one-third of its base area by its height—the most direct way to measure any pyramid's capacity.Area of a CircleThe area enclosed by a circle, calculated using its radius; needed for geometry and real-world problems involving circles like pools, wheels, or pizza.Pythagorean TheoremFind the length of any side of a right triangle given the other two sides, or check whether a triangle is a right triangle.

Frequently asked questions

How is the volume of a cone different from the volume of a cylinder?: A cone and a cylinder with the same radius and height have different volumes because the cone tapers to a point. Specifically, the cone's volume is exactly one-third of the cylinder's volume—hence the $\frac{1}{3}$ in the formula.
What do I do if I'm only given the slant height?: The slant height is the distance along the cone's surface from the base edge to the tip, not the perpendicular height used in the formula. Use the Pythagorean theorem with $h^2 + r^2 = l^2$ (where $l$ is slant height) to find the perpendicular height $h$ .
Can the volume of a cone be negative?: No, volume is always positive. Since $r$ , $h$ , and $\pi$ are all positive, $V = \frac{1}{3}\pi r^2 h$ must be positive. If you get a negative answer, check that you used positive values for radius and height.
How do I find the radius if I only know the volume and height?: Rearrange the formula to solve for $r$ : $V = \frac{1}{3}\pi r^2 h$ becomes $r^2 = \frac{3V}{\pi h}$ , then $r = \sqrt{\frac{3V}{\pi h}}$ . Make sure to take the positive square root since radius is always positive.

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Reviewed 2026-09-18