# Decagon Calculator

Welcome to the **Omni's decagon calculator**, where you'll learn all there is about those fascinating geometric shapes. Besides solving your homework problems about the **decagon side length, area, and perimeter**, it can also compute the length of all the diagonals and radii. As with every Omni tool, this decagon calculator is accompanied by a short article, where you find all **decagon-related formulas** as well as answers to some of the most burning questions that involve decagons:

- How to calculate the
**exterior angle**of a decagon? - How many
**diagonals**does a decagon have? - What is the
**apothem**of a decagon?

...and a few more. Let's go!

## Decagon perimeter, area, radii - formulas

Let's consider a regular decagon with side length * a*.

Since there are ten sides, it is quite obvious that the decagon perimeter formula reads:

**Perimeter = 10 × a**

The decagon area is slightly more complicated:

**Area = 2.5 × a² × √(5 + 2 × √5) ≈ 7.694 × a²**

*More information on how to calculate the area of a decagon awaits you in Omni's
decagon area calculator!*

The circumcircle radius * R* and the incircle radius

*can be computed as follows:*

**r***R* = ½ *a* × (1 + √5) ≈ 1.618 × *a*

*r* = ½ *a* × √ (5 + 2 × √5) ≈ 1.539 × *a*

## What are the angles in a decagon?

The decagon interior angle **α** is **144°**. Again, since there are ten interior angles, it follows easily that the sum of the interior angles of a decagon is **1440°**.

The exterior angle, **β**, is **36°** because it complements the interior angle to the straight angle, i.e., **α + β = 180°**.

Clearly, the angles do not depend in any way on the side length!

## How many diagonals does a decagon have?

A decagon has **35 diagonals**. Every diagonal spans across two, three, four, or five sides. More precisely, in a decagon of side * a* there are:

- Five diagonals spanning 5 sides, their length is
;*a*× (1 + √5) ≈ 3.236 ×*a* - Ten diagonals spanning 4 sides, their length is
;*a*× √(5 + 2 * √5) ≈ 3.078 ×*a* - Ten diagonals spanning 3 sides, their length is
**½**; and*a*× √(14 + 6 × √5) ≈ 2.618 ×*a* - Ten diagonals spanning 2 sides, their length is
**½**.*a*× √(10 + 2 × √5) ≈ 1.902 ×*a*

In the image below, each of the red and blue diagonals spans two sides, short black diagonals span three sides and long black diagonals span five sides. Can you see where the diagonals spanning four sides would fit in this image?

## How to use this decagon calculator?

Our decagon calculator can determine **all the properties of your decagon based on whatever data you have** - it need not be the side length! It can be the perimeter, area, radius (be it incircle or circumcircle), or any of the diagonals...

So input whatever you have, and the missing fields of the decagon calculator will get filled in immediately. You can also adjust the units to your needs. Enjoy!

## Similar Omni calculators

Satisfied with the decagon calculator? Since polygons are such a popular and fascinating topic, Omni features a whole **collection of polygon-related tools**! We bet you'll enjoy visiting:

## FAQ

### What is the apothem of a decagon?

The apothem of a regular decagon (and any other regular polygon) is the segment that joins the **center of the decagon with the midpoint of any of its sides**. It gives, therefore, the shortest distance possible from the center to the side. As such, it coincides with the **radius of the inscribed circle**.

### How do I calculate the exterior angle of a decagon?

To determine the exterior angle of a regular decagon:

- Recall the formula for the
**sum of interior angles**:`n × 180° - 360°`

, where`n`

is the number of sides. In our case`n = 10`

, so`10 × 180° - 360° = 1440°`

. - There are
**ten equal interior angles**, and their sum is`1440°`

, so each has`144°`

. - Recall that the
**exterior angle complements an interior angle**to`180°`

; that is:`144° + exterior = 180°`

. - It remains to conclude that
`exterior = 180° - 144° = 36°`

. Well done!

### What is the perimeter of a decagon with a side length of 2 in?

The perimeter is **20 in**, and the area is **30.777 in sq**. To get the perimeter, you just need to multiply the decagon side length by the number of sides, i.e., `10 × 2 in = 20 in`

. The area formula is more complicated, but approximately we have `Area ≈ 7.694 × a²`

, so, in our case, `Area ≈ 7.694 ×4 in sq ≈ 30.777 in sq`

.