Pi Day Logo

Octagon Calculator

Go back to Calculators page

The calculator is easy to use. Simply enter in the known values and the calculator will quickly give you the results you need. The perimeter, area, length of diagonals, as well as the radius of an inscribed circle and circumscribed circle will all be available in the blink of an eye.

What is a regular octagon? 

A regular octagon is a geometric shape with 8 equal lengths and 8 equal angles. The sum of the interior angles of a regular octagon is 1080 degrees, which makes each angle equal to 135 degrees in measure.

Area of a regular octagon:

One can think of the regular octagon as a square with corners that have been cut off or shortened. If you label “a” as the length of one side of an octagon, then the sides of the large square are \(a\sqrt{2}+1\).  From there, you can determine the area of the large square by squaring it \(a^2(3+2\sqrt{2})\).

The area of a octagon is the area of the 4 small triangles (\(a^2\)) formed from the truncation of the square, subtracted from the area of the square (S).

\(A=S-a^2=(3+2\sqrt{2})a^2-a^2\)

 

Perimeter

The perimeter is the easiest to calculate it’s just the sum of the lengths of all the sides.  Therefore, the perimeter, noted with a P is = 8a.

Suppose though that  you don’t know the length of an edge, but you know the area. The perimeter is determined by the formula:

\(P = \sqrt{\frac{32A}{\sqrt{2}+1}}\)

 

Diagonals

With numerous diagonals, the task of calculating the lengths may seem daunting. First, there are three different types of diagonals; we’ll call them “short”, “medium” (which is also known as the height of the octagon) and “long”.

The formulas for calculating the lengths are actually quite easy to use.

Short = \(a(\sqrt{2}+\sqrt{2})\)

Medium = \(a(\sqrt{2} +1)\)

Long = \(a(\sqrt{4}+2\sqrt{2})\)

Cicrumradius and Inradius

Simply put, the circumradius is half of the length of the longest diagonal, whereas the inradius is half of the height of the octagon.