Decagon Area Calculator

Created by Kenneth Alambra
Last updated: Aug 25, 2022

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.

Keep on reading to discover:

  • How to find the area of a decagon;
  • How to use this decagon area calculator; and
  • Some frequently asked questions about decagon.

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:

A=12×n×a×r\small A = \frac{1}{2}\times n\times a\times r

Where:

  • AA is the area of a regular polygon;
  • nn is the regular polygon's number of sides;
  • aa is the length of the regular polygon's side;
  • rr is the incircle radius or apothem of the regular polygon

Since we want to find the area of a decagon, we can replace nn with 1010. Consequently, we can find the value of rr using aa and nn with the help of this equation:

r=atan(πn)×2\small r = \frac{a}{\tan(\frac{\pi}{n})\times 2}

That said, we can simplify the regular polygon's area formula by replacing nn with 1010 and rr with that function to have:

Adecagon=12×n×a×r=12×n×a×atan(πn)×2=12×10×a2tan(180°10)×2=5× a2tan(18°)×2=2.5×a2tan(18°)\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 Adecagon=7.694208843×a2A_\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 cm5\ \text{cm}. Using our decagon area formula, we get:

Adecagon=2.5×a2tan(18°)=2.5×(5 cm)2tan(18°)=192.3552211 cm2192.36 cm2\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×a10\times a), circumcircle radius (R) (equal to a2×sin(18°)\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
regular polygon with sides, angles,circumcircle and incircle radii marked.








Name
regular decagon
α
144
deg
β
36
deg
Input at least one measurement
Side (a)
in
Perimeter
in
Circumcircle radius (R)
in
Incircle radius/ apothem (r)
in
Result
Area
in²
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