Cone Volume Calculator — Volume, Slant Height & Area

Cone formulas

Measure	Formula	For r = 3, h = 4
Slant height	s = √(r² + h²)	5
Volume	V = πr²h ÷ 3	37.6991 (12π)
Base area	B = πr²	28.2743 (9π)
Side (lateral) area	L = πrs	47.1239 (15π)
Total surface area	A = πr² + πrs	75.3982 (24π)
Angle at the base	atan(h ÷ r)	53.13°

The volume of any cone is one third of the cylinder with the same base and height — fill a cone three times and you fill the cylinder. The slant height is the distance down the side from the tip to the rim; the radius, height and slant height form a right triangle, which is why the 3-4-5 triangle gives such tidy numbers. The calculator accepts any two of radius or diameter, height, slant height and volume, and works out the rest. For the triangle on its own, the same √(a² + b²) rule applies.

Working backwards

Height from radius and slant height: h = √(s² − r²). A cone with r = 3 and s = 5 is 4 tall.
Height from radius and volume: h = 3V ÷ πr². A volume of 100 with r = 4 needs h = 5.96831.
Radius from height and volume: r = √(3V ÷ πh).

The slant height has to be longer than the radius and the height; if it is not, the measures cannot belong to one cone, and the calculator says so.

Piles of gravel, sand and mulch

A loose pile dumped from a truck or a conveyor forms a cone. To estimate how much is in it, measure the width across the base and the height to the top: a pile 10 feet across and 4 feet high is π × 5² × 4 ÷ 3 = 104.72 cubic feet, or 3.88 cubic yards. Real piles are rarely perfect cones, so treat the answer as an estimate. To order material by the cubic yard for a bed or a driveway, use the gravel calculator or the cubic yards calculator.

Cups, funnels and paper cones

A paper snow-cone cup 3 inches across and 4 inches deep holds 9.42 cubic inches, about 5.2 fluid ounces; the calculator shows the capacity in liters and gallons for any cone. To make a cone from flat paper or sheet metal, cut a circle sector whose radius is the slant height and whose angle is 360° × r ÷ s: for the 3-4-5 cone that is 216°, so you cut away 144° of a 5-inch circle. The side area — 47.12 square inches here — is the paper you need, without overlap for the seam. A party hat 6 inches across and 7 inches tall needs 71.8 square inches.

Cones, cylinders and spheres

A cone, a half sphere and a cylinder with the same radius and height equal to that radius have volumes in the ratio 1 : 2 : 3. For the other two solids, use the cylinder volume calculator and the sphere calculator. Results are rounded to six significant digits and also given in terms of π; the angles are in degrees.

Frequently asked questions

What is the formula for the volume of a cone?

V = πr²h ÷ 3, a third of the cylinder with the same base and height. A radius of 3 and a height of 4 give π × 9 × 4 ÷ 3 = 12π, or 37.6991 cubic units.

How do I find the slant height of a cone?

Use the Pythagorean theorem on the radius and height: s = √(r² + h²). For r = 3 and h = 4, s = √25 = 5.

What is the surface area of a cone?

The base plus the side: A = πr² + πrs. For r = 3 and s = 5 that is 9π + 15π = 24π, or 75.3982 square units.

How do I find the height from the volume?

h = 3V ÷ πr². A volume of 100 with a radius of 4 gives 300 ÷ 50.2655 = 5.96831.

How much gravel is in a cone-shaped pile?

Measure the width across the bottom and the height, then use V = πr²h ÷ 3. A pile 10 feet across and 4 feet high is 104.72 cubic feet, or 3.88 cubic yards.

What angle does a cone make?

The angle between the axis and the side is atan(r ÷ h). For r = 3 and h = 4 it is 36.87°, so the opening across the tip is 73.74° and the side meets the base at 53.13°.