Volume of Spheres, Cones, and Pyramids: Formulas and Worked Problems
October 21, 2026
Volume formulas for spheres, cones, and pyramids can be remembered with a simple pattern. A prism or cylinder has volume equal to base area times height. A cone or pyramid with the same base and height holds exactly one third as much. A sphere's volume is four thirds of π times the radius cubed. These show up in practical problems like calculating a pile of gravel, the capacity of a spherical tank, or the amount of ice cream in a cone.
Here are the formulas with worked problems.
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V = (4/3) × π × r³. A sphere with radius 3 cm has V = (4/3) × π × 27 = 113.1 cm³. A basketball with a radius of about 12 cm holds about 7,238 cm³ (7.2 liters). Because volume depends on the cube of the radius, doubling the radius multiplies volume by eight. A spherical water tank 10 feet in diameter (radius 5 ft) holds (4/3) × π × 125 = 523.6 cubic feet, or about 3,917 gallons.
Surface area of a sphere is 4πr², which is useful for paint or insulation estimates.
Cone
V = (1/3) × π × r² × h. A cone with radius 3 and height 8 has V = (1/3) × π × 9 × 8 = 75.4 cubic units. Compare with a cylinder of the same size: π × 9 × 8 = 226.2. The cone is exactly one third of it. A conical pile of sand 6 feet across at the base and 3 feet tall holds (1/3) × π × 9 × 3 = 28.3 cubic feet, about 1.05 cubic yards.
The slant height of a cone is √(r² + h²). For r = 3 and h = 8, it is 8.54, which matters for the surface area of a funnel or party hat.
Pyramid
V = (1/3) × base area × height. A pyramid with a 6 × 6 base and a height of 9 has V = (1/3) × 36 × 9 = 108 cubic units. The Great Pyramid of Giza originally had a base about 230 meters on a side and a height of about 147 meters, giving a volume of roughly (1/3) × 230² × 147 = 2.6 million cubic meters.
The same idea works for any base shape: triangular, hexagonal, or irregular. Compute the base area and use one third of base times height.
| Solid | Formula | Example | Volume |
|---|---|---|---|
| Sphere | (4/3)πr³ | r = 3 | 113.1 |
| Cone | (1/3)πr²h | r = 3, h = 8 | 75.4 |
| Pyramid | (1/3) × base × h | 6 x 6 base, h = 9 | 108.0 |
| Cylinder (for comparison) | πr²h | r = 3, h = 8 | 226.2 |
Why One Third?
Archimedes and earlier mathematicians showed that three pyramids of equal base and height can be assembled into a prism. In calculus the result follows from integrating the cross-sectional area, which grows with the square of the distance from the apex. Seeing this helps you remember that a cone is not half a cylinder, a common mistake.
Practical Problems
Gravel, sand, and salt are stored in cone-shaped piles. To estimate a pile's volume, measure the diameter and the height. Multiply by the material density to find weight. Grain bins and funnels, ice cream cones, and tents are all modeled by these formulas. For a hemisphere (half a sphere), divide by 2: a dome of radius 4 m has volume (2/3)π × 64 = 134 m³.
Frequently Asked Questions
What is the volume formula for a sphere?
(4/3) × π × r³.
Why is the volume of a cone one third of a cylinder?
Because the cross-section shrinks toward the tip, and integrating it gives one third of base area times height.
How do I find the volume of a hemisphere?
Half of a sphere's volume: (2/3) × π × r³.
How much does doubling the radius change a sphere's volume?
It multiplies volume by eight.
How do I find the volume of a sand pile?
Treat it as a cone, measuring the base diameter and height.