Volume of shapes: four formulas for box, cylinder, sphere and cone

Volume is how much space a solid takes up, and for the four shapes people actually run into — boxes, cylinders, spheres, cones — the formulas all come from one idea: the area of the base, extended through the height. The differences between them are exactly where and how that extension gets adjusted.

The method

For any shape with a constant cross-section swept straight up — a box or a cylinder — volume is simply base area × height. A box's base is a rectangle (l × w), so its volume is l × w × h. A cylinder's base is a circle (πr²), so its volume is πr² × h.

A cone tapers to a point instead of keeping a constant cross-section, and that taper is worth exactly one-third: ⅓ × πr² × h. (The same one-third shows up for any pyramid tapering to a point, not just a circular one.) A sphere has no flat base or straight height at all; its formula is its own thing, &frac43; × πr³.

A real example: a shipping box, two ways

A box measures 24 × 18 × 12 inches.

1. Volume: 24 × 18 × 12 = 5,184 cubic inches.
2. Cubic feet, since 1 cubic foot is 12 × 12 × 12 = 1,728 cubic inches: 5,184 ÷ 1,728 = 3 cubic feet, exactly.

That box happens to hold precisely 3 cubic feet — a clean number worth checking by hand before trusting a freight quote that bills by the cubic foot.

A real example: a cylindrical tank

A tank is a cylinder with a 3-inch radius and a 10-inch height.

1. Base area: π × 3² = π × 9.
2. Volume: 9 × π × 10 = 90π ≈ 282.743339 cubic inches.

Same idea as the box — base area times height — just with a circular base instead of a rectangular one.

A real example: a sphere

A ball has a 4-inch radius.

1. r³ = 4³ = 64.
2. Volume: &frac43; × π × 64 = 85.333333 × π ≈ 268.082573 cubic inches.

There's no "height" input here at all — a sphere's radius alone determines the whole shape, which is why its formula looks different from the other three.

A real example: a cone

A cone-shaped hopper has a 6-inch radius and a 9-inch height.

1. Base area: π × 6² = 36π.
2. Full cylinder of the same base and height would be 36π × 9 = 324π.
3. The cone is one-third of that: 324π ÷ 3 = 108π ≈ 339.292007 cubic inches.

Worth noticing: a cone with the exact same radius and height as a cylinder holds exactly one-third as much — not an approximation, an exact ratio every time.

Where people get tripped up

Doubling a radius doesn't double the volume — it multiplies it by four for a circle's area and by eight for a sphere or cylinder's volume, because radius appears squared or cubed in every one of these formulas. A tank with twice the radius and the same height doesn't hold twice as much liquid; it holds four times as much. That's the single most common estimating mistake with round containers.

Run your own numbers

The free Volume of Shapes calculator switches between box, cylinder, sphere and cone, showing only the measurements each one needs, entirely on your device.