Surface Area Calculator

Q: What is the surface area formula for a sphere?

$SA = 4\pi r²$, where r is the radius. A sphere with radius 5 m has a surface area of about 314.16 m².

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

Choose a shape from the list of seven options.
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
The result appears automatically once every required field has a value.
Read the surface area in square units. A copy button next to the result copies it to the clipboard.

While a required field is still blank the calculator shows a short prompt instead of a result. Once every field has a value, a number that is zero or negative produces an error message.

For a pyramid, one extra rule applies to every base, square or rectangular: the slant height entered must be longer than half the base width. Half the base width is the distance from the center of the base to the middle of a base-length edge. A slant height equal to or shorter than that leaves no room for any vertical height, so no such pyramid exists and the calculator returns an error.

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: $SA = 4\pi r²$
Cube: $SA = 6a²$ , where a is the side length
Cylinder: $SA = 2\pi r² + 2\pi rh$ , the area of the two circular ends plus the curved side
Cone: $SA = \pi r² + \pi rs$ , where the slant height $s = \sqrt{r² + h²}$
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 $a = c = \sqrt{h² + (b/2)²}$ for a symmetrical (isosceles) triangle

A pyramid with a rectangular base has two different slant heights, one for each pair of triangular faces. The Slant Height box takes the slant height s of the two faces that stand on the base-length edges. From it the calculator recovers the vertical height h with $h = \sqrt{s² - (w/2)²}$ , then finds the other pair of faces:

Pyramid with a rectangular base: $SA = lw + ls + w\sqrt{h² + (l/2)²}$ , where l is the base length, w is the base width, and h is the vertical height derived from s.

The square-base formula above is the special case of this one, reached when the base length and base width are equal (to within 0.001 of the entered units).

Worked example

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-base pyramid, base length 6 m, base width 8 m, slant height 5 m: half the base width is 4 m, so the vertical height is √(5² − 4²) = 3 m. The other pair of faces has slant height √(3² + 3²) = √18 ≈ 4.2426 m. SA = (6 × 8) + (6 × 5) + (8 × 4.2426) = 48 + 30 + 33.94 ≈ 111.94 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 ≈ 4.272 m. SA = (3 × 4) + (4.272 + 3 + 4.272) × 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? $SA = 4\pi r²$ , 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: $SA = 2\pi r² + 2\pi rh$ , 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. Enter a base length and a base width that differ, plus the slant height of the two faces standing on the base-length edges. The calculator derives the vertical height and the second slant height from those three numbers.

Why does the pyramid sometimes show an error? The slant height must be greater than half the base width. At or below that value the pyramid would have zero or imaginary height, so no such solid exists and no surface area can be reported.

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

References

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

Surface Area Calculator - Free Tool for All 3D Shapes