Area & Volume Calculator

Shape & Dimensions

Use any consistent unit — results follow whatever you put in.

Shape

Radius

—

Unit label (display only)

PRIMARY RESULT

—

Area—

Perimeter—

——

Area scales with the square of a length and volume with the cube. Doubling the radius of a sphere multiplies its volume by eight, not by two.

Share & embed

Units are not optional

Geometry formulas do not care what unit you use, but they do care that you use the same one throughout. Mixing centimetres and metres in a single calculation is the most common source of a wrong answer, and it produces a result that is wrong by a factor of a hundred or ten thousand rather than by a little — so it is usually obvious once you look at the magnitude.

Area comes out in squared units and volume in cubed units. If you enter metres, an area is in square metres and a volume in cubic metres. The unit selector here is a label only; it never converts anything behind your back.

Heron's formula for awkward triangles

When you know three sides but no height, the base-times-height formula is useless. Heron's formula computes the area from the three sides alone using the semi-perimeter. It also provides a free validity check: if the three lengths cannot form a triangle — if one side is longer than the other two combined — the formula produces a negative value under the square root, and the shape is flagged as impossible rather than silently returning nonsense.

The square-cube law

Doubling every dimension of a shape multiplies its area by four and its volume by eight. This is why large animals have proportionally thicker legs, why a scaled-up model of a bridge is not a valid test of the real one, and why a storage tank twice as tall and twice as wide holds eight times as much while needing only four times the material. It is the most practically useful consequence of geometry and it surprises people every time.

Slant height versus vertical height on cones

A cone has two different "heights": the vertical height from base to apex, and the slant height along the sloping surface. Volume uses the vertical height; lateral surface area uses the slant height. Using the wrong one is the classic cone mistake. This calculator asks for the vertical height and derives the slant height from it using Pythagoras, so both results stay consistent.

Frequently Asked Questions

Do I need to convert units before entering dimensions?

Only to make them consistent with each other. The formulas work in any unit, but all dimensions in one calculation must use the same one. Areas come out squared and volumes cubed in whatever unit you supplied.

How do I find a triangle's area when I only know the three sides?

Use Heron's formula, which works from the sides alone via the semi-perimeter. Choose the three-sides triangle option. It also detects impossible triangles, where one side exceeds the sum of the other two.

If I double a shape's size, what happens to area and volume?

Area multiplies by four and volume by eight. This square-cube relationship is why scale models do not behave like the full-size object and why large structures need proportionally more material.

Should a cone's volume use the slant height?

No. Volume uses the vertical height from base to apex; only the lateral surface area uses the slant height. This calculator asks for the vertical height and derives the slant height itself to keep both consistent.