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!

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 r can be computed as follows:

R = ½ a × (1 + √5) ≈ 1.618 × a

r = ½ a × √ (5 + 2 × √5) ≈ 1.539 × a

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!

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 a × √(14 + 6 × √5) ≈ 2.618 × a; and
• 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?

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!

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:

• Heptagon calculator;
• Heptagon area calculator;
• Dodecagon calculator;
• Dodecagon area calculator;
• Polygon angle calculator; and
• Polygon calculator.

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.

To determine the exterior angle of a regular decagon:

1. 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°.
2. There are ten equal interior angles, and their sum is 1440°, so each has 144°.
3. Recall that the exterior angle complements an interior angle to 180°; that is: 144° + exterior = 180°.
4. It remains to conclude that exterior = 180° - 144° = 36°. Well done!

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.