Surface Area Calculator (With Steps) — Calculate Surface Area of a Cone, Cube, Cylinder, Sphere & More

This surface area calculator lets you calculate the surface area for several different solid shapes. The shapes currently supported are cones, cubes, cuboids (rectangular prisms), cylinders, and spheres.

For most shapes, you have the option to choose what combination of inputs you want to provide. And to top it all, the calculator will give you not just the answer, but also the step by step solution.

Here's a quick overview of the concept of surface area and its formula for a few important shapes.

Surface Area — Concept and Formulas

The surface area of any solid or three-dimensional object is a measure of how large or small its surface is.

Surface Area of a Cone

A cone can be seen has two surfaces — a curved surface and a flat circular base.

A labeled diagram of a cone showing its surfaces — A cone

For a cone with a radius $\hspace{0.2em} r \hspace{0.2em}$ and slant height $\hspace{0.2em} l \hspace{0.2em}$ , the curved surface area of a cone is $\hspace{0.2em} \pi r l \hspace{0.2em}$ . And that of the base is $\hspace{0.2em} 2 \pi r \hspace{0.2em}$ .

The total surface area is the sum of these two area. So,

\begin{align*} S \hspace{0.25em} &= \hspace{0.25em} \pi r^2 + 2 \pi r l \\[1em] &= \hspace{0.25em} \pi r (r + l) \end{align*}

Surface Area of a Cube

A cube has six flat surfaces with equal areas.

A cube with an edge length of a — A cube

So, for a cube with an edge-length of $\hspace{0.2em} a \hspace{0.2em}$ , each flat surface has an area of $\hspace{0.2em} a^2 \hspace{0.2em}$ and the total surface area would be,

S = 6 a^2

Surface Area of a Cuboid (Rectangular Prism)

A cuboid has three pairs of parallel, opposite, and equal flat surfaces. So, its surface area is the sum of the areas of these three pairs.

A cuboid with an length of l, breadth b, and height h — A cuboid/rectangular prism

The formula for the surface area of a cuboid (or rectangular prism) with a length $\hspace{0.2em} l \hspace{0.2em}$ , width $\hspace{0.2em} w \hspace{0.2em}$ , height $\hspace{0.2em} h \hspace{0.2em}$ is

S = 2 \hspace{0.05em} (lw + wh + lh)

Surface Area of a Cylinder

A cylinder has a curved surface and two flat circular bases.

A labeled diagrab of a cylinder showing its surfaces — A cylinder

So the surface area of a cylinder with a radius $\hspace{0.2em} r \hspace{0.2em}$ and height $\hspace{0.2em} h \hspace{0.2em}$ is

\begin{align*} S \hspace{0.25em} &= \hspace{0.25em} 2 \pi r^2 + 2 \pi r h \\[1em] &= \hspace{0.25em} 2 \pi r (r + h) \end{align*}

Surface Area of a Sphere

The sphere of radius $\hspace{0.2em} r \hspace{0.2em}$ has a surface area equal to

S = 4 \pi r^2

Surface Area Calculations — Examples

Example

What is the surface area of a cone with a radius of 5 centimeters and a slant height of 7 centimeters.

Solution

Using the formula for a cone's surface area, we have

\begin{align*} A \hspace{0.25em} &= \hspace{0.25em} \pi \cdot r \cdot (r + l) \\[1em] &= \hspace{0.25em} \pi \cdot 5 \cdot (5 + 7) \\[1em] &= \hspace{0.25em} 188.49 \end{align*}

The surface area of the cone is $\hspace{0.2em} 188.49 \text{ cm}^2 \hspace{0.2em}$ .

Example

A cylindrical container has a radius of 2 inches and a height of 6 inches. Calculate its total surface area.

Solution

The surface area of a cylinder is given by

S \hspace{0.25em} = \hspace{0.25em} 2 \pi r (r + h)

Substituting the values of $\hspace{0.2em} r \hspace{0.2em}$ and $\hspace{0.2em} h \hspace{0.2em}$ , we get

\begin{align*} S \hspace{0.25em} &= \hspace{0.25em} 2 \pi r (r + h) \\[1em] &= \hspace{0.25em} 2 \pi \cdot 2 \cdot (2 + 6) \\[1em] &= \hspace{0.25em} 100.53 \end{align*}

The total surface area of the cylinder is $\hspace{0.2em} 100.53 \text{ in}^2 \hspace{0.2em}$ .

Example

The surface area of a sphere is $\hspace{0.2em} 144 \pi \text{ in}^2 \hspace{0.2em}$ . Find its radius.

Solution

We know the surface area of a sphere is given by the formula

A \hspace{0.25em} = \hspace{0.25em} 4 \pi r^2

Substituting the values of $\hspace{0.2em} A \hspace{0.2em}$ and solving for $\hspace{0.2em} r \hspace{0.2em}$ —

\begin{align*} 144 \pi \hspace{0.25em} &= \hspace{0.25em} 4 \pi r^2 \\[1em] r \hspace{0.25em} &= \hspace{0.25em} 6 \end{align*}

So, the radius of the sphere is $\hspace{0.2em} 6 \text{ in} \hspace{0.2em}$ .

Example

The surface area of a rectangular prism is $\hspace{0.2em} 184 \text{ cm}^2 \hspace{0.2em}$ . Its length is $\hspace{0.2em} 8 \text{ cm} \hspace{0.2em}$ and width $\hspace{0.2em} 5 \text{ cm} \hspace{0.2em}$ . Find the height of the prism.

Solution

The surface area of a rectangular prism is

A \hspace{0.25em} = \hspace{0.25em} 2 \cdot (lw + wh + lh)

Substituting the values of $\hspace{0.2em} l \hspace{0.2em}$ , $\hspace{0.2em} w \hspace{0.2em}$ , and $\hspace{0.2em} A \hspace{0.2em}$ and solving for $\hspace{0.2em} h \hspace{0.2em}$ , we have

\begin{align*} 184 \hspace{0.25em} &= \hspace{0.25em} 2 \cdot (8 \cdot 5 + 5 \cdot h + 8 \cdot h) \\[1em] 184 \hspace{0.25em} &= \hspace{0.25em} 2 \cdot (40 + 5h + 8h) \\[1em] 184 \hspace{0.25em} &= \hspace{0.25em} 80 + 26h \\[1em] h \hspace{0.25em} &= \hspace{0.25em} 4 \end{align*}

That's it. The prism's height is $\hspace{0.2em} 4 \text{ cm} \hspace{0.2em}$ .

Surface Area Calculator

About the Surface Area Calculator

Usage Guide

i. Valid Inputs

ii. Example

iii. Solutions

iv. Share

Surface Area — Concept and Formulas

Surface Area of a Cone

Surface Area of a Cube

Surface Area of a Cuboid (Rectangular Prism)

Surface Area of a Cylinder

Surface Area of a Sphere

Surface Area Calculations — Examples