Surface Area Calculator - Free Tool for All 3D Shapes

Surface Area Calculator

The surface area of a three-dimensional shape is the total area of every face and curved surface that covers its outside, measured in square units such as square meters or square feet. This calculator finds the surface area of seven common solids: sphere, cube, cylinder, pyramid, cone, rectangular prism, and triangular prism.

What surface area means

Surface area is the sum of the areas of all the outer surfaces of a solid. A cube's surface area is the combined area of its six flat square faces. A sphere's surface area is the area of its single curved outer surface. Surface area is different from volume, which measures the space inside a solid rather than the area of its boundary. Both are measured in different units: surface area in square units (m²), volume in cubic units (m³).

How to calculate surface area

1. Choose a shape from the list of seven options.
2. Enter the measurements the shape needs:
   • Sphere: radius
   • Cube: side length
   • Cylinder: radius and height
   • Pyramid: base length, base width, and slant height
   • Cone: radius and height
   • Rectangular prism: length, width, and height
   • Triangular prism: base length, height of the triangular face, and length of the prism
3. The result appears automatically once every required field has a value.
4. Read the surface area in square units. A copy button next to the result copies it to the clipboard.

If any field is left blank or holds a number that is zero or negative, the calculator shows an error instead of a result.

For a pyramid, an extra rule applies when the base is a rectangle rather than a square: the slant height entered must be longer than half the base width. Below that value no real pyramid with those proportions exists, so the calculator returns an error. This rule does not apply to a square base, which is solved with a simpler formula directly.

Surface area formulas

In the formulas below, r is radius, h is height, l is length, w is width, b is base length, and s is slant height.

• Sphere: SA = 4πr²
• Cube: SA = 6s², where s is the side length
• Cylinder: SA = 2πr² + 2πrh, the area of the two circular ends plus the curved side
• Cone: SA = πr² + πrs, where the slant height s = √(r² + h²)
• Pyramid with a square base: SA = l² + 2ls, where l is the base length and s is the slant height of a triangular face
• Rectangular prism: SA = 2(lw + lh + wh)
• Triangular prism (triangular cross-section with base b, height h, and two equal slant sides a = c): SA = bh + (a + b + c)L, where L is the length of the prism and a = c = √(h² + (b/2)²) for a symmetrical (isosceles) triangle

A pyramid with a rectangular base, where the base length and base width differ, has two different slant heights, one for the pair of faces on the long edges and one for the pair on the short edges. The calculator works this out from the base length, base width, and the slant height of one pair of faces. The square-base formula above is the special case where both slant heights are equal.

Worked examples

Sphere, radius 5 m: SA = 4π(5)² = 100π ≈ 314.16 m²

Cube, side length 3 m: SA = 6(3)² = 54 m²

Cylinder, radius 2 m, height 5 m: SA = 2π(2)(2 + 5) = 28π ≈ 87.96 m²

Cone, radius 3 m, height 4 m: slant height = √(3² + 4²) = 5 m, so SA = π(3)(3 + 5) = 24π ≈ 75.40 m²

Square-base pyramid, base length 4 m, slant height 5 m: SA = 4² + 2(4)(5) = 16 + 40 = 56 m²

Rectangular prism, length 4 m, width 3 m, height 5 m: SA = 2(4·3 + 4·5 + 3·5) = 2(12 + 20 + 15) = 94 m²

Triangular prism, base 3 m, triangle height 4 m, prism length 5 m: the two equal slant sides are √(4² + 1.5²) = √18.25 ≈ 4.27 m. SA = (3 × 4) + (4.27 + 3 + 4.27) × 5 = 12 + 57.72 ≈ 69.72 m²

Units and precision

All input measurements should use the same unit, such as meters, feet, or centimeters. The result is given in that unit squared. The calculator computes with full floating-point precision internally and displays the result rounded to two decimal places.

Why surface area matters

Surface area is used to estimate the paint or coating needed for a wall or object, the material needed for packaging, and the sheet metal or fabric needed to build a container. In biology and chemistry, the surface area of a cell or a catalyst particle affects how fast it exchanges material with its surroundings, since exchange happens across the surface. In engineering, the surface area of a radiator or heat exchanger affects how quickly it can transfer heat.

Frequently asked questions

What is surface area? Surface area is the total area of every outer face or curved surface of a three-dimensional object, given in square units.

What is the surface area formula for a sphere? SA = 4πr², where r is the radius. A sphere with radius 5 m has a surface area of about 314.16 m².

How do you find the surface area of a cylinder? Add the area of the two circular ends to the area of the curved side: SA = 2πr² + 2πrh, where r is the radius and h is the height.

What is the difference between surface area and volume? Surface area measures the outer boundary of a shape in square units. Volume measures the space enclosed by that boundary in cubic units. A shape's surface area and volume usually grow at different rates as its size increases.

Can this calculator handle a pyramid with a rectangular base? Yes. Entering a base length and base width that differ tells the calculator to treat the pyramid as rectangular rather than square, and it solves for the two different slant heights automatically.

What shapes does this calculator support? Sphere, cube, cylinder, pyramid, cone, rectangular prism, and triangular prism.

References

1. "Surface Area." Wikipedia, Wikimedia Foundation. https://en.wikipedia.org/wiki/Surface_area
2. Weisstein, Eric W. "Surface Area." MathWorld — A Wolfram Web Resource. https://mathworld.wolfram.com/SurfaceArea.html
3. "Surface Area Formulas." Math Is Fun. https://www.mathsisfun.com/geometry/surface-area.html