Cone Volume Calculator

What is a cone?

A cone is a 3D shape with a flat circular base and one pointed tip, called the apex. Its side narrows in a smooth curve from the base to the apex. Ice cream cones, traffic cones, funnels and party hats are all cones. In geometry, "cone" usually means a right circular cone: the apex sits straight above the center of the base.

Parts of a cone

• Apex: the pointed tip.
• Base: the flat circle at the bottom.
• Radius (r): the distance from the center of the base to its edge.
• Height (h): the straight distance from the apex down to the base.
• Slant height (s): the distance along the side, from the apex to the edge of the base.
• Axis: the line from the apex to the center of the base. A cone is a solid of revolution. Spin a right triangle around one of its short sides, and it sweeps out a cone.

Right cone vs oblique cone

In a right cone, the apex is straight above the center of the base. In an oblique cone, the apex leans to one side. Both have the same volume if they have the same base and the same perpendicular height. Always measure the height straight up from the base, not along the side.

Cone volume formula

The formula for the volume of a cone is:

• V is the volume.
• π (pi) is about 3.14159.
• r is the radius of the base.
• h is the height. In words: find the area of the base circle (π × r²), multiply it by the height, then divide by 3.

Volume of a cone with diameter

If you know the diameter instead of the radius, use this version:

Example: A cone 8 inches across and 9 inches tall holds 3.14159 × 64 × 9 ÷ 12 = 150.8 cubic inches.

Cone volume with the base area

For any cone, with any shape of base, the volume is one third of the base area times the height:

This is the same rule as for a pyramid. A cone is like a pyramid with a round base.

How to find the volume of a cone

To calculate the volume of a cone by hand:

1. Measure the radius of the base. If you have the diameter, divide it by 2.
2. Square the radius: multiply it by itself.
3. Multiply by π (3.14159) to get the base area.
4. Multiply by the height.
5. Divide by 3. The answer is in cubic units, such as cubic inches or cubic centimeters.

Cone volume example in inches

A funnel has a radius of 3 inches and a height of 4 inches.

1. r² = 3 × 3 = 9.
2. Base area = 3.14159 × 9 = 28.27 square inches.
3. 28.27 × 4 = 113.1.
4. 113.1 ÷ 3 = 37.7 cubic inches. The funnel holds about 0.16 gallons, or 0.62 liters.

Cone volume example in centimeters

An ice cream cone has a radius of 2.5 cm and a height of 15 cm. V = 1/3 × 3.14159 × 2.5² × 15 = 98.17 cm³. That is almost 100 mL.

Cone volume chart

Use this chart to check your answer. Each row loads into the cone volume calculator.

Compare the rows for 2 × 4 and 12 × 24 inches. Making a cone 6 times bigger in every direction makes its volume 216 times bigger (6 × 6 × 6).

A cone with a volume of exactly π

A cone with a radius of 1 and a height of 3 has V = 1/3 × π × 1 × 3 = π, about 3.14159. This is an easy way to check that you have the formula right.

Why a cone is one third of a cylinder

A cone holds exactly one third of a cylinder with the same base and height. Fill the cone with water three times and pour it into the cylinder, and the cylinder is full.

See the cylinder volume calculator for the full shape.

Cone dimensions: height, radius and slant height

The radius, the height and the slant height of a cone form a right triangle. If you know two of them, you can find the third with the Pythagorean theorem.

Example: A cone with a radius of 3 inches and a slant height of 5 inches has a height of √(25 − 9) = 4 inches.

How to find the height of a cone

If you know the volume and the radius, divide 3 times the volume by the base area:

Example: A cone holds 100 cubic inches and has a radius of 3 inches. h = 300 ÷ (3.14159 × 9) = 10.61 inches. If you know the slant height instead, use h = √(s² − r²).

How to find the radius of a cone

If you know the volume and the height:

Example: A cone 8 inches tall that holds 1 gallon (231 cubic inches) has a radius of √(693 ÷ 25.13) = 5.25 inches, so it is 10.5 inches across.

Volume of a cone with the slant height

Find the height first, then use the volume formula. Example: A cone with a radius of 6 inches and a slant height of 10 inches has h = √(100 − 36) = 8 inches. V = 1/3 × 3.14159 × 36 × 8 = 301.6 cubic inches.

How to find the slant height of a cone

Add the squares of the radius and the height, then take the square root.

Example: A cone with a radius of 4 inches and a height of 9 inches has a slant height of √(16 + 81) = 9.85 inches. You need the slant height for the surface area and for the flat pattern of the cone.

Cone angles

Looking at a cone from the side, you see a triangle. Its angles describe how pointed the cone is.

Surface area of a cone

The surface area of a cone has two parts: the flat base and the curved side, called the lateral surface.

Example: A cone with r = 3 and s = 5 inches has a base of 28.27 square inches and a lateral surface of 3.14159 × 3 × 5 = 47.12 square inches. The total surface area of the cone is 75.4 square inches.

How to find the surface area of a cone

1. Find the slant height: s = √(r² + h²).
2. Multiply π × r × s to get the lateral area.
3. Add π × r² for the base.

For an open cone with no base, such as a funnel or a party hat, use the lateral area only.

Lateral surface area of a cone: a party hat example

A party hat is a cone with no base, so you only need its curved side. A hat 7 inches across and 7 inches tall has r = 3.5 and h = 7. Its slant height is √(12.25 + 49) = 7.83 inches. The paper it needs is π × 3.5 × 7.83 = 86.05 square inches, plus a little extra for the overlap you glue.

Cone surface area from the height

If you know the height but not the slant height:

Example: An ice cream cone with r = 1.5 and h = 5 inches has a slant height of 5.22 inches. Its side, the part made of waffle, has an area of 3.14159 × 1.5 × 5.22 = 24.6 square inches.

Cone net: unrolling the side of a cone

Cut a paper cone along its side and lay it flat, and the side becomes a slice of a circle, called a sector. The sector's radius is the slant height s.

Example: For a cone with r = 3 and s = 5, the sector angle is 360° × 3 ÷ 5 = 216°. Cut out a 216° slice from a circle with a 5-inch radius, roll it up, and you get the cone. This is how party hats, lamp shades and metal funnels are made. The calculator shows the sector angle in the details.

Truncated cone (frustum) volume

A truncated cone, or frustum, is a cone with the top cut off by a flat cut parallel to the base. Flower pots, buckets, lamp shades and drinking cups are often frustums.

R is the radius of the bottom, r is the radius of the top, and h is the height between them.

Example: A planter 12 inches across at the bottom, 9 inches across at the top and 8 inches tall has R = 6 and r = 4.5. V = 1/3 × 3.14159 × 8 × (36 + 27 + 20.25) = 697.4 cubic inches, or about 3 gallons of soil.

You can also find a frustum as a big cone minus the small cone cut off its top. For more on frustums, use the truncated cone calculator.

Volume of liquid in a cone

A cone gets wider as you go up, so the liquid level does not rise evenly. A cone glass filled to half its height holds only one eighth of its full volume.

To fill a point-down cone halfway by volume, fill it to 79.4% of its height.

Marking a cone glass for equal parts

Because the cone is narrow at the bottom, the marks for equal volumes are not evenly spaced. For a cone with its point down, put the marks at these heights:

This is why marks on conical measuring cups, cocktail shakers and funnels get closer together near the top. Example: A cone hopper 48 inches across and 36 inches tall holds 94 gallons when full. With grain or water 18 inches deep, it holds only 11.75 gallons.

For a tank with a cone bottom, use the tank volume calculator.

Ice cream cone and snow cone volume

An ice cream cone or snow cone with a round top is a cone plus a half ball (a hemisphere). Add the two volumes.

Example: A cone with a 1.5-inch radius and a height of 4 inches holds 9.42 cubic inches. A round scoop on top adds 7.07 cubic inches. The total is 16.49 cubic inches, or about 9.1 fluid ounces.

Sizes vary by brand. Measure your own cone for the best answer.

Volume of a conical pile

A pile of sand, gravel, mulch or grain forms a cone. Its sides settle at a slope that depends on the material and how wet it is. If you know the width of the pile and its slope angle, you can find the height and the volume:

Example: A sand pile 10 feet across with a 34° slope is 5 × tan 34° = 3.37 feet tall. Its volume is 1/3 × 3.14159 × 25 × 3.37 = 88.3 cubic feet, or 3.27 cubic yards.

If you can measure the height directly, use the cone mode with the radius and the height instead. If the pile is too wide to measure across, walk a tape around its base. Divide the distance around by 6.2832 to get the radius. A pile 31.4 feet around has a radius of 5 feet. To turn the volume into weight, ask your supplier for the weight per cubic yard of the material. For bulk orders, see the cubic feet calculator.

How to measure a cone

You need the radius and the height. Here is how to measure each one on a real object.

• Radius: measure straight across the open end or the base, through the center, to get the diameter. Divide by 2.
• Height: stand the cone on its base. Lay a ruler flat across the tip and measure straight down to the table. Do not measure along the side.
• Slant height: run a tape along the side from the tip to the rim. The calculator can find the height from it.
• Inside or outside: to find how much a cone holds, measure the inside. A thick waffle cone or a funnel wall takes up space.
• Use water to check: fill the cone with water and pour it into a measuring cup. Compare the amount with the calculator's answer in milliliters or fluid ounces.

Cone vs cylinder, sphere and pyramid

The cone shares its "one third" rule with the pyramid, and links neatly to the cylinder and the sphere.

So a cone, a sphere and a cylinder with the same radius and a height of 2r have volumes in the ratio 1 : 2 : 3. See the sphere volume calculator for more.

Cone volume in gallons, liters and cubic feet

Example: A conical tank bottom 4 feet across and 3 feet deep holds 1/3 × 3.14159 × 2² × 3 = 12.57 cubic feet, or 94 gallons. For more unit changes, use the gallon converter.

Cone volume with calculus

Slice a cone into thin discs from the apex (z = 0) down to the base (z = h). At height z, the disc's radius is r × z ÷ h. Add up all the discs:

The disc's radius grows in a straight line, so its area grows with z². That z² becomes z³ ÷ 3 when you add the slices, and that is where the 1/3 comes from.

Common cone volume mistakes

• Forgetting to divide by 3. Without the 1/3, you get the volume of a cylinder.
• Using the diameter as the radius. This makes the answer 4 times too big.
• Using the slant height as the height. Find the height with h = √(s² − r²) first.
• Mixing units. Keep the radius and the height in the same unit.
• Treating a cup or a pot as a cone. If it has a flat bottom, it is a frustum. Use the truncated cone mode.
• Assuming a half-full cone is half the volume. A point-down cone filled to half its height holds only 12.5%.

Where cone volume is used

• Kitchen: funnels, ice cream cones, piping bags and conical strainers.
• Construction and landscaping: piles of sand, gravel and mulch.
• Farming and industry: grain hoppers, silos with cone bottoms and cyclone dust collectors.
• Science: test tubes, conical flasks and centrifuge tubes.
• Safety and events: traffic cones and party hats.
• Design: lamp shades, speakers and paper cups.

Sources

We checked the formulas and values on this page against these sources.

1. Cone — Wolfram MathWorld. “Cone volume, slant height and surface area.”
2. Conical Frustum — Wolfram MathWorld. “Truncated cone volume and surface area.”
3. Cavalieri's Principle — Wolfram MathWorld. “Why an oblique cone has the same volume as a right cone.”
4. NIST Handbook 44, Appendix C — National Institute of Standards and Technology. “Cubic inch, gallon and liter values.”