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# Decagon Area Calculator

How to find the area of a decagonHow to use this calculatorOther related calculatorsFAQs

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).

This tool also has an extra feature where you can determine the area of other common regular polygons. Simply tick the Find other polygon's area checkbox below the Diagram and angles section of our decagon area calculator to display the Number of sides variable and enter another whole number to find that polygon's area.

FAQs

### 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