📏 Enter known values

📐 Accepted units: mm, cm, dm, m, dam, hm, km, in, ft, yd, mi

Formula Reference

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Worked example

Data
Radius
\(b = {\color{#6c5ce7}{3\,\text{cm}}}\)
Height
\(c = {\color{#6c5ce7}{5\,\text{cm}}}\)
Volume
\(a = \,?\)
Formula
\[a = \frac{\pi\cdot{{b}}^{2}\cdot{c}}{3}\]
Steps
1Substitute the values
\[a = \frac{\pi\cdot{\left({\color{#6c5ce7}{3\,\text{cm}}}\right)}^{2}\cdot{\color{#6c5ce7}{5\,\text{cm}}}}{3}\]
Each letter is replaced by its value.
2Square
\[a = \frac{\pi\cdot9\,\text{cm}^{2}\cdot{\color{#6c5ce7}{5\,\text{cm}}}}{3}\]
\(3 \times 3 = 9\)\(\text{cm} \times \text{cm} = \text{cm}^{2}\)
3Multiply
\[a \approx \frac{141.3717\,\text{cm}^{3}}{3}\]
\(\pi\cdot9 \approx 28.2743\)\(28.2743\cdot5 \approx 141.3717\)\(\text{cm}^{2} \times \text{cm} = \text{cm}^{3}\)
4Divide
\[a \approx \frac{141.3717\,\text{cm}^{3}}{3} \approx 47.1239\,\text{cm}^{3}\]
\(\frac{141.3717}{3} \approx 47.1239\)
Volume: 47.1239 cm³
Check

If we use the result to calculate Radius, we should get the value you entered:

\[b = \sqrt{\frac{3\cdot{a}}{\pi\cdot{c}}}\]
\[b = \sqrt{\frac{3\cdot47.1239\,\text{cm}^{3}}{\pi\cdot5\,\text{cm}}} \approx 3\,\text{cm}\]
It gives 3 cm, the same as the value entered.
Calculate the Volume
Please fill in the fields:
Radius Height
And leave empty
Volume
Calculate the Radius
Please fill in the fields:
Volume Height
And leave empty
Radius
Calculate the Height
Please fill in the fields:
Volume Radius
And leave empty
Height

How to calculate the volume of a cone

A cone holds exactly one third of the cylinder with the same base and height. Its base is a circle, so:

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

From the volume you can get the height or the radius:

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

\[ r = \sqrt{\frac{3V}{\pi h}} \]

The height is the perpendicular distance from the tip to the base, not the length measured along the side (the slant height).

How to use the calculator

  1. Enter two of the three values: the volume, the radius and the height.
  2. Leave empty the one you want to find.
  3. Press Calculate. The volume comes out in cubic units: cm³, m³.

Worked examples

1. Radius of 3 cm and height of 4 cm

\[ V = \frac{1}{3} \pi \times 3^2 \times 4 = 12\pi \approx 37.7\ \text{cm}^3 \]

2. Ice cream cone

A cone 12 cm tall with an opening 5 cm across (a radius of 2.5 cm) holds \( \frac{1}{3}\pi \times 2.5^2 \times 12 = 25\pi \approx 78.54\ \text{cm}^3 \), about 79 mL.

3. A pile of sand

A conical pile of sand is 1.5 m tall and its base is 4 m across. Its volume is \( \frac{1}{3}\pi \times 2^2 \times 1.5 = 2\pi \approx 6.28\ \text{m}^3 \).

4. Height from the volume

A cone of 500 cm³ with a radius of 5 cm is \( 3 \times 500 \div (\pi \times 25) \approx 19.1 \) cm tall.

Slant height and height

If you measured along the side of the cone (the slant height \( g \)), find the height with the Pythagorean theorem:

\[ h = \sqrt{g^2 - r^2} \]

Example: a slant height of 13 and a radius of 5 give a height of \( \sqrt{169 - 25} = 12 \).

Volume of cones with a height of 10

Volume units: 1 cm³ = 1 mL, 1 dm³ = 1 L and 1 m³ = 1,000 L.

RadiusBase areaVolume
13.1410.47
212.5741.89
328.2794.25
450.27167.55
578.54261.8
6113.1376.99
8201.06670.21
10314.161,047.2

When Do You Need to Calculate the Volume of a Cone?

🍦 Ice Cream and Food

Ice cream cones, piping bags and paper cups are cones. Their volume tells how much they hold, which matters for portions and prices.

Portion sizes and recipe planning
🏗️ Piles of Sand, Gravel and Grain

Loose material poured on the ground forms a cone. Measuring its height and its base gives the volume of the pile without weighing it.

Estimate stockpiles on site
🌾 Silos and Hoppers

Silos and hoppers have a conical bottom. Its volume is added to the cylindrical part to get the total capacity.

Total capacity of storage bins
🧪 Labs and Funnels

Funnels and centrifuge tubes have conical sections. The volume of the cone gives the capacity and the graduation marks.

Measure liquids in conical containers
🎉 Party Hats and Crafts

Paper cones for hats, popcorn or candy hold a volume set by their radius and height. Designers use it to size them.

Size conical containers for crafts and events
📚 Geometry Exams

The cone is the classic example of the factor 1/3, and exercises often give the slant height to test the Pythagorean theorem.

Practice with volume and right triangles

Common Mistakes

⚠️ Forgetting to Divide by 3
Common Error: Using πr²h, which is the volume of the cylinder. The cone holds only one third of it.
✅ Solution: V = πr²h ÷ 3
⚠️ Using the Slant Height
Common Error: Measuring along the side and using that length as the height. It is longer than the real height, so the volume comes out too large.
✅ Solution: h = √(slant height² − r²)
⚠️ Using the Diameter as the Radius
Common Error: Entering the diameter in the radius field gives a volume four times too large.
✅ Solution: Radius = diameter ÷ 2
⚠️ Not Squaring the Radius
Common Error: Writing π × r × h ÷ 3.
✅ Solution: Square the radius first
⚠️ Wrong Units
Common Error: Giving the volume in cm² or mixing centimeters and meters. Volume is in cubic units.
✅ Solution: Use cm³, m³ or liters (1 L = 1,000 cm³)
⚠️ Truncated Cones
Common Error: Using the cone formula for a bucket or a lampshade, which is a cone with the tip cut off (a frustum).
✅ Solution: For a frustum: V = πh(R² + Rr + r²) ÷ 3

Applications by Industry

Construction & Mining
  • Stockpiles: Volume of sand, gravel and ore piles
  • Foundations: Concrete for conical footings
  • Roofs: Air volume under conical roofs for ventilation
Agriculture
  • Grain piles: Volume of grain heaped on the floor
  • Hoppers: Capacity of the conical bottom of silos
  • Spreaders: Hopper capacity of fertilizer spreaders
Science & Labs
  • Funnels: Capacity of filter funnels
  • Tubes: Graduation of conical centrifuge tubes
  • Geology: Volume of volcanic cones
Food Industry
  • Ice cream: Capacity of cones and portions
  • Pastry: Piping bag fillings
  • Packaging: Conical cups and paper cones
Manufacturing
  • Molds: Material for conical molds and parts
  • Machining: Material removed when turning a cone
  • Speakers: Air volume in loudspeaker cones
Education
  • Geometry: Why a cone is one third of a cylinder
  • Calculus: Volume of solids of revolution
  • Exams: Problems with the slant height

External Resources

Other calculators

Volume of a Pyramid
Volume of a Pyramid
Volume of a Square Prism
Volume of a Square Prism
Volume of a Cube
Volume of a Cube
Volume of a Cylinder
Volume of a Cylinder
Volume of a Sphere
Volume of a Sphere
Volume of a Rectangular Prism
Volume of a Rectangular Prism
Area of a Square
Area of a Square
Area of a Circle
Area of a Circle

Geometry Calculators

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