Surface Area Calculator

Surface area is the total area of all the faces, sides, and curved surfaces of a 3D object. Think of it as the amount of wrapping paper you'd need to cover the shape completely, with no gaps and no overlap.

The basic process is the same for most shapes: identify every face or surface, calculate the area of each one, then add them all together. For flat-faced shapes like cubes and prisms, that means working with rectangles, squares, and triangles. Curved shapes like spheres and cylinders have their own specific formulas, but the idea is identical.

A few things to keep in mind before you start:

• Make sure all your measurements use the same unit (inches, centimeters, feet, etc.) before plugging them in.
• Surface area is always expressed in square units, like cm² or ft².
• Some shapes have a total surface area (all surfaces) and a lateral surface area (just the sides, excluding the top and bottom). Know which one you need.

Here's a quick reference for the shapes covered on this page. Each formula is explained in more detail in its own section below.

These formulas cover the most common scenarios. Variables like r (radius), h (height), l (length or slant height), and a (side length) will show up repeatedly, so it's worth getting comfortable with them.

A cube is the simplest case. All six faces are identical squares, so calculating its surface area is pretty straightforward once you know the side length.

If you have a cube with side length a, the total surface area is:

SA = 6a²

Just square the side length and multiply by 6. That's it. For example, a cube with sides of 4 cm has a surface area of 6 × 16 = 96 cm².

A sphere has no flat faces or edges, just one continuous curved surface. That makes the formula a little different from the others, but it's still simple enough to work with.

SA = 4πr²

Here, r is the radius of the sphere (the distance from the center to any point on the surface). If you only know the diameter, just divide it by 2 to get the radius first.

Example: a sphere with a radius of 5 cm has a surface area of 4 × π × 25 ≈ 314.16 cm².

A cylinder has two circular ends and one curved side that wraps around them. To find the total surface area, you add up all three parts: the top circle, the bottom circle, and the curved lateral surface in between.

Total SA = 2πr² + 2πrh

• r = radius of the circular base
• h = height of the cylinder

For a cylinder with radius 3 cm and height 10 cm: SA = 2π(9) + 2π(3)(10) = 56.55 + 188.50 ≈ 245.04 cm².

A cone has a circular base and a curved surface that tapers up to a point. The total surface area adds those two parts together.

Total SA = πr² + πrl

• r = radius of the circular base
• l = slant height (the distance along the side of the cone from the base edge to the tip)

Example: a cone with radius 4 cm and slant height 9 cm has a total surface area of π(16) + π(4)(9) = 50.27 + 113.10 ≈ 163.36 cm².

If you only know the vertical height of the cone rather than the slant height, you can find the slant height using the Pythagorean theorem: l = √(r² + h²).

A rectangular prism is basically a box: six rectangular faces, three pairs of opposite sides that are identical. The total surface area adds up all six faces.

SA = 2(lw + lh + wh)

• l = length
• w = width
• h = height

For a box that's 5 cm long, 3 cm wide, and 4 cm tall: SA = 2(15 + 20 + 12) = 2(47) = 94 cm².

This formula works because there are three pairs of faces. One pair has area l × w, another has l × h, and the third has w × h. Multiply each pair's area by 2 (since there are two of each), then add everything together.

A triangular prism has two triangular ends and three rectangular faces connecting them. The total surface area is the sum of both triangles plus all three rectangles.

SA = bh_t + (s₁ + s₂ + s₃) × l

• b = base of the triangle
• h_t = height of the triangle (the perpendicular height, not the prism's length)
• s₁, s₂, s₃ = the three side lengths of the triangular face
• l = length (depth) of the prism

The two triangular ends each have area ½ × b × h_t, so together they contribute b × h_t. Each rectangular face has area equal to one side of the triangle multiplied by the prism's length. Add it all up and you have your total surface area.

For a triangular prism where the triangle has a base of 6 cm, a height of 4 cm, and sides of 5, 5, and 6 cm, with a prism length of 10 cm: SA = (6 × 4) + (5 + 5 + 6) × 10 = 24 + 160 = 184 cm².

The most common pyramid in math problems is the square pyramid: a square base with four triangular faces that meet at a point. The total surface area is the area of the base plus the area of all four triangular sides.

SA = b² + 2bl

• b = side length of the square base
• l = slant height of a triangular face

Each triangular face has an area of ½ × b × l. With four of them, that's 4 × ½ × b × l = 2bl. Add the square base (b²) and you get the total.

Example: a square pyramid with a base side of 8 m and a slant height of 10 m has a surface area of 64 + 2(8)(10) = 64 + 160 = 224 m².

For pyramids with other base shapes (triangular, hexagonal, etc.), the approach is the same: find the area of the base, calculate the area of each triangular face using the appropriate slant height, and add everything together.

Here's a consolidated reference chart for all the shapes covered above. Use this as a quick lookup when you know the shape and just need the formula.

Working through a few examples is the fastest way to get comfortable with these formulas. Here are some straightforward ones for the most common shapes.

Cube with side length 6 cm:
SA = 6 × 6² = 6 × 36 = 216 cm²

Sphere with radius 7 cm:
SA = 4π × 7² = 4π × 49 ≈ 615.75 cm²

Cylinder with radius 4 cm and height 12 cm:
SA = 2π(4²) + 2π(4)(12) = 100.53 + 301.59 ≈ 402.12 cm²

Cone with radius 5 cm and vertical height 12 cm:
First, find slant height: l = √(25 + 144) = √169 = 13 cm
SA = π(25) + π(5)(13) = 78.54 + 204.20 ≈ 282.74 cm²

Rectangular prism measuring 10 cm × 6 cm × 4 cm:
SA = 2(60 + 40 + 24) = 2(124) = 248 cm²

Each of these follows the same pattern: plug the measurements into the right formula, work through the arithmetic carefully, and don't forget to label your answer in square units.

This distinction comes up constantly, and it's worth being clear on what each term actually means.

Total surface area is the area of every surface on a shape, including the base(s) and any top or end caps. It represents the complete outer area of the object.

Lateral surface area covers only the sides of a shape, leaving out the top and bottom. For a cylinder, that's just the curved tube part. For a pyramid, it's the four triangular faces without the square base.

Which one you need depends entirely on the problem. A few common scenarios:

• Painting the outside of a closed box: use total surface area.
• Calculating the label area on a soup can (no top or bottom): use lateral surface area.
• Figuring out how much roofing material a pyramid-shaped roof needs (no floor): use lateral surface area.
• Wrapping a gift box completely: use total surface area.

When a problem doesn't specify, total surface area is usually the safer assumption. But if the context involves open containers, labels, or just the sides of a structure, lateral surface area is what you're after.