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Surface Area Calculator
A surface area calculator finds the total outside area of a shape. This one covers 15 shapes — cube, sphere, cylinder, rectangular prism, cone, square pyramid, triangular prism, regular tetrahedron, hemisphere, cone frustum, capsule, ellipsoid, torus, hollow cylinder, and rectangular pyramid. Pick a shape, enter its dimensions, and it works out the surface area in square units — the number you need for paint, wrapping paper, fabric, or any job that covers a surface.
| Shape | |
| Formula used | |
| Surface area |
How it works
Surface area is the total area of every face on the outside of a three-dimensional shape. Imagine peeling the wrapping paper off a gift box and laying it flat: the area of that flat sheet is the box's surface area. It is always measured in square units — square inches, square feet, or square centimeters — because it is an area, not a length or a volume. Pick the shape, enter the dimensions in the same unit, and the calculator applies the right formula.
A cube has six identical square faces, so its surface area is six times the area of one face: 6 × side². A sphere has a single curved surface with no flat faces at all, and its area is 4 × π × radius². A cylinder has two circular ends plus a curved side that unrolls into a rectangle, giving 2 × π × radius × (radius + height). A rectangular prism, or box, has three pairs of identical rectangular faces, so its area is 2 × (length × width + length × height + width × height).
A cone adds a circular base to a curved side that unrolls into a sector: π × radius × (radius + slant height), where the slant height is √(radius² + height²). A square pyramid has a square base plus four identical triangles: base² + 2 × base × slant height. A triangular prism has two triangular ends plus three rectangles: 2 × triangle area + perimeter × length. A regular tetrahedron is four identical equilateral triangles, so its area is simply √3 × edge². A hemisphere (a closed half-sphere, like a dome with a floor) has the curved half of a sphere plus a flat circular base: 3 × π × radius².
A cone frustum is a cone with its tip cut off — a bucket or lampshade shape — and its area is the slanted side plus both circular ends: π × (R + r) × slant + π × R² + π × r². A capsule is a cylinder with a hemisphere on each end: 2 × π × radius × (2 × radius + cylinder height). A torus, or donut shape, has area 4 × π² × R × r, where R is the distance from the center of the hole to the center of the tube and r is the tube radius. A hollow cylinder, like a pipe, has an outer side, an inner side, and two ring-shaped ends: 2 × π × (R + r) × height + 2 × π × (R² − r²). A rectangular pyramid has a rectangular base and four triangles of two different sizes: length × width + length × slant over the width + width × slant over the length. An ellipsoid is a stretched sphere; it has no exact formula using only basic functions, so the calculator uses the Knud Thomsen approximation (p = 1.6075, accurate to about 1%) and labels the result approximate.
Box: 2 × (l×w + l×h + w×h) • Cone: π × r × (r + s) • Square pyramid: a² + 2 × a × l
Triangular prism: 2 × A + P × L • Tetrahedron: √3 × a² • Hemisphere: 3 × π × r²
Frustum: π × (R + r) × s + π × R² + π × r² • Capsule: 2 × π × r × (2r + h)
Torus: 4 × π² × R × r • Hollow cylinder: 2 × π × (R + r) × h + 2 × π × (R² − r²)
Rectangular pyramid: l×w + l × lw + w × ll • Ellipsoid: Knud Thomsen approximation
All answers are in square units — the square of whatever unit you measured in
The most common mistake is mixing up surface area and volume. Volume is the space inside a shape, measured in cubic units, and it answers "how much fits in here?" Surface area answers "how much material covers the outside?" A shipping box needs cardboard for its surface area and space for its volume, and the two numbers behave differently: double every side of a cube and its surface area grows four times, while its volume grows eight times. That is why small objects feel light for their size and large ones need so much paint.
Surface area is a working number, not just a school exercise. A painter estimating a job measures the walls in square feet to know how many gallons of paint to buy. A gift wrapper checks whether a sheet of paper covers a box. A gardener building a raised bed works out the area of the boards. A baker knows that two small pizzas have more crust edge — and more surface area — than one large pizza of the same total size. In each case the shape's formula turns a few measurements into the exact amount of material needed.
Worked example
A woodworker in Nashville, Tennessee, is building a wooden cube-shaped storage box with a side length of 3 feet, and she wants to know how much varnish to buy to coat the entire outside.
A cube has six faces, each one a square with side 3 feet. Step 1 — area of one face: 3 × 3 = 9 square feet. Step 2 — multiply by six faces: 6 × 9 = 54 square feet. She needs enough varnish for 54 square feet of surface.
For a second check, take a sphere with radius 2: its surface area is 4 × π × 2² = 4 × 3.14159 × 4 = 50.27 square units. Select "Cube" with side 3 in the calculator above and you get 54; select "Sphere" with radius 2 and you get 50.27.
One more: a cone with radius 3 and height 4. Step 1 — slant height: √(3² + 4²) = √25 = 5. Step 2 — surface area: π × 3 × (3 + 5) = 24 × π = 75.40 square units. A torus with major radius 5 and minor radius 2 is even simpler: 4 × π² × 5 × 2 = 394.78 square units.
FAQ
What is surface area?
Surface area is the total area of all the outside faces of a three-dimensional shape, measured in square units such as square inches or square feet. It tells you how much material you need to cover the shape: paint for a wall, wrapping paper for a box, or fabric for a cushion.
What is the difference between surface area and volume?
Surface area measures the outside of a shape in square units, while volume measures the space inside it in cubic units. A box can have a large surface area but a small volume, or the other way around. Paint covers surface area; what fits inside the box is volume.
What is the surface area formula for a sphere?
The surface area of a sphere is 4 x pi x radius squared (4 x pi x r squared). A sphere with radius 2 has a surface area of 4 x 3.14159 x 4 = 50.27 square units. A sphere has no flat faces or edges, just one curved surface.
Does the cylinder formula include the top and bottom?
Yes, the total surface area of a closed cylinder includes both circular ends plus the curved side: 2 x pi x r x (r + h). If you need the area of an open-top cylinder, like a bucket with no lid, leave off one circle and use 2 x pi x r x h + pi x r squared instead.
How many shapes does this calculator support?
This calculator supports 15 shapes: cube, sphere, cylinder, rectangular prism, cone, square pyramid, triangular prism, regular tetrahedron, hemisphere, cone frustum, capsule, ellipsoid, torus, hollow cylinder, and rectangular pyramid. Pick the shape from the dropdown and the calculator shows the right inputs, formula, and diagram for it.
Why is the ellipsoid surface area approximate?
Unlike a sphere, an ellipsoid has no exact surface-area formula using only basic functions, so this calculator uses the Knud Thomsen approximation with p = 1.6075, which is accurate to about 1 percent for most ellipsoids. The result is labelled approximate. For a sphere (where all three axes are equal) the approximation gives the exact sphere result.