Surface Area Calculator

What this calculator does

Surface area is the total area of the outside of a three-dimensional shape, every face and curved surface added together. This one tool works it out for five common solids: a cone, a cube, a cylinder, a rectangular prism, and a sphere.

Pick the shape, type its measurements, and it adds up the outside for you.

What surface area actually is

The easiest way to picture it is wrapping paper. If you wanted to cover a shape completely, or paint every outside surface, the surface area is how much you would need to do it.

Because it measures area, it comes in square units, centimetres squared or metres squared, even though it is wrapped around a solid object. That is the thing to hold onto, and it is what separates surface area from volume. Volume is the space inside a shape, measured in cubic units. Surface area is the skin outside it, measured in square units. One fills the shape, the other covers it.

Using the calculator

1. Choose the shape from the dropdown. The input boxes change to match it.
2. Type the measurements it asks for, and pick a unit.
3. Press Calculate.

All measurements must be positive, and the result comes back in square units of the unit you choose.

The five shapes and their formulas

In every case the outside is just its surfaces added up:

• Cube: six equal square faces, 6 × edge².
• Rectangular prism: six rectangles in three matching pairs, 2 × (width × length + length × height + width × height).
• Sphere: 4 × π × radius².
• Cylinder: two circular ends plus the wrapped-around side, (2 × π × radius²) + (2 × π × radius × height).
• Cone: the circular base plus the slanted side, (π × radius²) + (π × radius × √(radius² + height²)), where √(radius² + height²) is the slant height.

Seeing it as a net

A clear way to understand surface area is to unfold the shape. Cut a cube or a box along enough of its edges and lay it out flat, and you get a net, a single flat pattern showing every face at once. A cube unfolds into six squares, a box into six rectangles. The surface area is nothing more than the total area of that flat net, which is why each formula is a sum of face areas.

Square units, and the π it uses

The answer is in square units, matched to the unit you pick and shown with a small 2, so a shape measured in centimetres gives a surface area in cm². Keep all measurements in one unit before calculating. For the three round shapes, the sphere, cylinder and cone, the tool uses π as 3.141592654, far finer than anything you would measure.

Two quick examples

A cube with an edge of 5 cm: six faces of 5 × 5, so 6 × 25 = 150 cm².

A sphere with a radius of 10 cm: 4 × π × 10² = 4 × π × 100, which is about 1,256.64 cm².

Going deeper on one shape

This tool is the all-in-one switchboard. For a single shape with its formula explained more fully, worked through step by step, there is a dedicated tool for each: the surface area of a cube calculator, the rectangular prism, the cylinder, the cone, and the sphere.

Questions people ask

What is surface area?

The total area of the outside of a solid, all its faces and curved surfaces added together. It is measured in square units.

Which shapes does this handle?

Five: the cone, cube, cylinder, rectangular prism and sphere. Pick one from the dropdown and the inputs adjust.

What units does the answer use?

Square units, like cm² or m², since surface area is an area even though it wraps a 3D shape.

How is surface area different from volume?

Surface area is the skin on the outside, in square units. Volume is the space on the inside, in cubic units. One covers the shape, the other fills it.

What value of π does it use?

For the round shapes it uses π as 3.141592654, ten significant figures.

References

A note on the idea behind it. The surface area of a solid is the sum of the areas of all its faces and curved surfaces, measured in square units, and is commonly found by unfolding the solid into a flat net. For the round shapes the formulas rest on the value of π tabulated by the US National Institute of Standards and Technology. For further reading, see Surface area.

1. Surface area, the total area of all the faces and curved surfaces of a solid, found by summing the face areas or unfolding the solid into a net.
2. National Institute of Standards and Technology (NIST), Digital Library of Mathematical Functions, value of π used for the round shapes. https://dlmf.nist.gov/