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About the Surface Area Calculator

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.

Usage Guide

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i. Valid Inputs

Each of the inputs provided must be a non-negative real number. In other words, the input must be 0 or greater. Here are a few examples.

If you would like to see an example of the calculator's working, just click the "example" button.

iii. Solutions

As mentioned earlier, the calculator won't just tell you the answer but also the steps you can follow to do the calculation yourself. The "show/hide solution" button would be available to you after the calculator has processed your input.

iv. Share

We would love to see you share our calculators with your family, friends, or anyone else who might find it useful.

By checking the "include calculation" checkbox, you can share your calculation as well.

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.

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.

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.

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.

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

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.

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