What if the cone is tilted (an oblique cone)?

The formula still works, as long as $h$ is the perpendicular height from the apex to the plane of the base. A tilted cone with the same base and the same perpendicular height has the same volume as a straight one.

When should I leave the answer in terms of $\pi$?

When the problem asks for an exact answer or says "in terms of $\pi$", write it like $48\pi$ cm³. When you need a number for a real situation, multiply by $3.14$ (or use the $\pi$ key) and round to the place the problem asks for.

How do I find the height if I only know the slant height?

The radius, the height and the slant height make a right triangle with the slant height as the hypotenuse. So $h = \sqrt{l^2 - r^2}$. For $r = 5$ and $l = 13$: $h = \sqrt{169 - 25} = \sqrt{144} = 12$.

Teacher Resources

Volume of Cones

Learn the cone volume formula V = (1/3)πr²h, why a cone holds one third of a cylinder, and how to use the perpendicular height.

Intermediate25 minLesson

Printable worksheet

Definition

A cone is a 3D shape with a flat circular base that narrows to a single point called the apex (or vertex). Think of an ice cream cone, a party hat or a traffic cone.

The Volume Formula

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

Where:

$r$ = radius of the circular base
$h$ = height, the perpendicular distance from the apex straight down to the center of the base
$\pi r^2$ = the area of the base

Why One Third?

In the cylinder lesson you learned

V = \pi r^2 h

. Take a cylinder and a cone with the same base and the same height. If you fill the cone with water and pour it into the cylinder, it takes exactly 3 full cones to fill the cylinder. So the cone holds one third of the cylinder:

V_{\text{cone}} = \frac{1}{3} \times V_{\text{cylinder}} = \frac{1}{3}\pi r^2 h

Careful: the height

h

is not the slant height

l

(the length along the sloping side). The radius, the height and the slant height form a right triangle, so

l^2 = r^2 + h^2

Try it now

Which formula gives the volume of a cone with radius

r

and height

h

Worked Examples

A waffle cone has a radius of 3 cm and a height of 10 cm. Find its volume in terms of $\pi$ , then with $\pi \approx 3.14$ , rounded to the nearest tenth.

Write the formula

$V = \frac{1}{3}\pi r^2 h$ → Formula ready

Square the radius

$r^2 = 3^2 = 9$ → $r^2 = 9$

Multiply by the height and divide by 3

$\frac{1}{3} \times 9 \times 10 = \frac{90}{3} = 30$ → $V = 30\pi$ cm³

Use $\pi \approx 3.14$

$30 \times 3.14 = 94.2$ → $V \approx 94.2$ cm³

Answer: The cone holds exactly $30\pi$ cm³, which is about $94.2$ cm³ of ice cream (not counting the scoop on top!).

Common Mistakes

Forgetting the $\frac{1}{3}$ and using $V = \pi r^2 h$

Why it's wrong: That is the cylinder formula. A cone with the same base and height holds only one third of the cylinder, so the answer comes out 3 times too big.

Correct: Always write $V = \frac{1}{3}\pi r^2 h$ and divide by 3. For $r = 3$ , $h = 10$ : $\frac{1}{3}\pi \times 9 \times 10 = 30\pi$ , not $90\pi$ .

Using the diameter instead of the radius

Why it's wrong: The radius is half the diameter. Squaring the diameter makes the answer 4 times too big.

Correct: If the problem gives the diameter, halve it first: a diameter of 8 cm means $r = 4$ cm and $r^2 = 16$ , not $64$ .

Using the slant height as the height

Why it's wrong: The slant height runs along the sloping side, so it is always longer than the perpendicular height. Using it makes the volume too big.

Correct: The height goes straight from the apex down to the center of the base. If you only know the slant height $l$ , find $h$ with $h^2 = l^2 - r^2$ .

Writing the answer in square units, like cm²

Why it's wrong: Volume is three-dimensional: $r \times r \times h$ multiplies three lengths.

Correct: Volume is always in cubic units: cm³, m³, in³. Remember that $1$ cm³ $= 1$ mL.

Interactive Visual

3D Shape Viewer

Select Shape

Rotate X (20°)

Rotate Y (30°)

Radius (3)

Height (10)

Faces

Edges

Vertices

Volume

V = ⅓πr²h

V = ⅓π · 3² · 10 = 30π ≈ 94.25

94.25 cm³

Surface Area

SA = πr² + πrl

SA = π · 3² + π · 3 · 10.44 ≈ 126.67

126.67 cm²

l = √(3² + 10²) ≈ 10.44

Interactive Sandbox

Expression Calculator

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History

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Practice Problems

18 problems

Problem 1 of 18

Easy

Which formula gives the volume of a cone with radius $r$ and height $h$ ?

Why It Matters

Cones are all around you, and knowing their volume answers real questions:

Food: How much ice cream fits in a waffle cone? How much popcorn does a paper cone hold?
Kitchen and garage: How much liquid sits in a funnel before it drains?
Construction: How many cubic meters of sand, gravel or road salt are in a cone-shaped pile?
Industry: Grain silos and hoppers often have a cone-shaped bottom, so engineers need cone volumes to know their capacity.
Nature: Volcanoes and anthills are roughly cone-shaped, so scientists estimate their volume with this formula.

The one third rule also comes back in the pyramid lesson, so learning it here gives you two formulas for the price of one!

Real World Applications

Ice Cream Shops

Ice cream shops choose cone sizes and prices by how much ice cream fits inside each cone.

Example:

A small cone with radius 2 cm and height 9 cm holds $\frac{1}{3}\pi \times 4 \times 9 = 12\pi \approx 37.68$ cm³.

1Try It Yourself

A large waffle cone has a radius of 3 cm and a height of 12 cm. The shop fills it right up to the rim.

How many cubic centimeters of ice cream fit inside? Use 3.14 for $\pi$ .

Step 1: Write the mathematical expression

Calculate $\frac{1}{3} \times 3.14 \times 3^2 \times 12$ :

Paper Cups at a Water Dispenser

Many office water dispensers use cone-shaped paper cups. They are cheap to make and easy to stack.

Example:

A paper cone cup with radius 3 cm and height 10 cm holds $30\pi \approx 94.2$ cm³, which is about $94$ mL of water.

2Try It Yourself

A cone-shaped paper cup has a radius of 4 cm and a height of 9 cm.

How many milliliters of water does it hold? Use 3.14 for $\pi$ (1 cm³ = 1 mL).

Step 1: Write the mathematical expression

Calculate $\frac{1}{3} \times 3.14 \times 4^2 \times 9$ :

Piles of Sand and Road Salt

When sand, gravel or road salt is poured from above, it forms a cone. Builders and road crews use the cone formula to estimate how much material is in a pile.

Example:

A sand pile with radius 3 m and height 2 m contains $6\pi \approx 18.84$ m³ of sand.

3Try It Yourself

A pile of road salt for the winter forms a cone with a diameter of 10 m and a height of 3 m.

How many cubic meters of salt are in the pile? Use 3.14 for $\pi$ .

Step 1: Write the mathematical expression

Halve the diameter, then calculate $\frac{1}{3} \times 3.14 \times r^2 \times 3$ :

Key Takeaways

1The volume of a cone is $V = \frac{1}{3}\pi r^2 h$ , one third of base area times height
2A cone holds exactly one third of a cylinder with the same base and the same height
3Use the radius, not the diameter: halve the diameter before squaring
4The height $h$ is the perpendicular height, not the slant height; if you know the slant height $l$ , use $h^2 = l^2 - r^2$
5Give exact answers in terms of $\pi$ (like $30\pi$ cm³) or approximate ones with $\pi \approx 3.14$ , rounded as the problem asks
6To find a missing height, use $h = \frac{3V}{\pi r^2}$ ; to find a missing radius, use $r^2 = \frac{3V}{\pi h}$ and take the square root

Frequently Asked Questions

A cone gets narrower all the way up, so most of its volume sits near the base. You can test it with water or rice: three full cones fill a cylinder with the same base and height. Later, in calculus, you can prove that the factor is exactly

\frac{1}{3}

. Pyramids follow the same one third rule.

Glossary

Cone: A 3D shape with a circular base and curved sides that meet at one point, the apex
Apex: The pointed tip of a cone (also called the vertex)
Radius: The distance from the center of the circular base to its edge, half the diameter
Height (perpendicular height): The straight-line distance from the apex to the center of the base, at a right angle to the base
Slant height: The distance from the apex to the edge of the base along the sloping side; $l^2 = r^2 + h^2$
In terms of $\pi$: An exact answer that keeps the symbol $\pi$ , like $30\pi$ cm³, instead of a rounded decimal

Formula Card

Volume of a cone

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

One third of base area ($\pi r^2$) times perpendicular height

Cone and cylinder

V_{\text{cone}} = \frac{1}{3} V_{\text{cylinder}}

Same base and same height: three cones fill one cylinder

Missing height

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

Multiply the volume by 3, then divide by the base area

Height from slant height

h = \sqrt{l^2 - r^2}

Pythagorean theorem in the right triangle formed by $r$, $h$ and $l$

Volume of Cones

Definition

The Volume Formula

Why One Third?

Worked Examples

Common Mistakes

Interactive Visual

3D Shape Viewer

Interactive Sandbox

Expression Calculator

History

Practice Problems

Why It Matters

Real World Applications

Ice Cream Shops

Paper Cups at a Water Dispenser

Piles of Sand and Road Salt

Key Takeaways

Frequently Asked Questions

Glossary

Formula Card

Volume of a cone

Cone and cylinder

Missing height

Height from slant height

More in This Topic

Introduction to Volume

Metric Volume Units (mL, L)

Volume of Rectangular Prisms

Imperial Volume Units (cups, pints, gallons)