# Decagon Area Calculator

Created by Kenneth Alambra
Last updated: Jan 20, 2023

This decagon area calculator will help you quickly determine the area of a regular decagon. A decagon is a regular polygon with 10 sides. These 10 sides are equal in measurement and comprise the perimeter of the decagon. Like any polygon, a decagon has an enclosed space or area that we can calculate using geometry and trigonometry.

• How to find the area of a decagon;
• How to use this decagon area calculator; and

## How to find the area of a decagon

To find the area of a decagon, we can use the area formula of any regular polygon, as shown below:

$\small A = \frac{1}{2}\times n\times a\times r$

Where:

• $A$ is the area of a regular polygon;
• $n$ is the regular polygon's number of sides;
• $a$ is the length of the regular polygon's side;
• $r$ is the incircle radius or apothem of the regular polygon

Since we want to find the area of a decagon, we can replace $n$ with $10$. Consequently, we can find the value of $r$ using $a$ and $n$ with the help of this equation:

$\small r = \frac{a}{\tan(\frac{\pi}{n})\times 2}$

That said, we can simplify the regular polygon's area formula by replacing $n$ with $10$ and $r$ with that function to have:

\small \begin{align*} A_\text{decagon} &= \frac{1}{2}\times n\times a\times r\\\\ &= \frac{1}{2}\times n\times a\times \frac{a}{\tan(\frac{\pi}{n})\times 2}\\\\ &= \frac{1}{2}\times 10\times \frac{a^2}{\tan(\frac{180\degree}{10})\times 2}\\\\ &= 5\times \ \frac{a^2}{\tan(18\degree)\times 2}\\\\ &= \frac{2.5\times a^2}{\tan(18\degree)}\\\\ \end{align*}

We can simplify it further to $A_\text{decagon} = 7.694208843\times a^2$ but isn't it nice to know how we derived this area of a regular decagon formula? 🙂

Now, let's say we have a decagon with a side that measures $5\ \text{cm}$. Using our decagon area formula, we get:

\small \begin{align*} A_\text{decagon} &= \frac{2.5\times a^2}{\tan(18\degree)}\\\\ &= \frac{2.5\times (5\ \text{cm})^2}{\tan(18\degree)}\\\\ &= 192.3552211\ \text{cm}^2\\ &≈ 192.36\ \text{cm}^2 \end{align*}

## How to use this calculator

This tool is very straightforward. To find the area of a regular decagon, all you have to do is:

• Enter the measurement of side (a); or
• Input other measurements of your decagon like its perimeter (which is only $10\times a$), circumcircle radius (R) (equal to $\frac{a}{2\times \sin(18\degree)}$), or the apothem (r).

Finding this tool interesting? Here are some of our other polygon tools that you might find interesting too:

## FAQ

### How do I calculate decagon area?

To calculate the decagon area:

1. Measure the side of the decagon. Let's say we get 2 inches.
2. Square the side of the decagon. 2 inches × 2 inches = 4 in².
3. Multiply the squared side by 2.5/tan(18°) or 7.694208843 to have: 4 in² × 7.694208843 = 30.77683537 in² ≈ 30.777 in².

### What is the area of a decagon with a side of 3 cm?

The area of a 3-cm-sided decagon is approximately 69.25 cm². We get that value by substituting 3 cm as the value of a in this formula:

decagon area = 7.694208843 × a².

Substituting 3 cm, we get:
decagon area = 7.694208843 × (3 cm)²
= 7.694208843 × 9 cm²
= 69.24787959 cm²
≈ 69.25 cm²

Kenneth Alambra

Name
regular decagon
α
144
deg
β
36
deg
Input at least one measurement
Side (a)
in
Perimeter
in
in
in
Result
Area
in²
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