Octagon calculator
Online calculator and formulas for a regular octagon
This function calculates various parameters of an octagon.
To perform the calculation, select the required parameter in the menu and enter its value. Then click the 'Calculate' button.
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Properties of an octagon
A regular octagon is a polygon consisting of eight corners and eight sides. If all eight sides are the same length, we speak of an equilateral octagon. Even if all the angles at the eight corners are the same size, the octagon is called a regular octagon.
For a regular octagon, the diagonals that connect opposite vertices are all the same length and all interior angles of the regular hexagon are equal and 135 degrees.
The sum of the interior angles of a regular octagon is 1080° (6 x 180°). This comes from a general formula for polygons, in which the number of vertices of the polygon is used as a variable \(n\):
\((n-2)· 180\ \ \ = (8-2) · 180 \ \ = 6 · 180 =1080°\)
The value of an interior angle of 135° is therefore given by the formula:
\(\displaystyle \frac{180·(n-2)}{n} \ \ =\frac{180·(8-2)}{8} =135° \)
These two formulas apply to all regular polygons. The number of corners is used for n.
Formulas for the regular octagon
Perimeter (\(\small{P}\))
\(\displaystyle P = a · 8 \)
Area (\(\small{A}\))
\(\displaystyle A = a^2 · 2 · (1+\sqrt{2}) \) \(\displaystyle \ \ ≈a^2 · 4.8284 \)
\(\displaystyle A = 2 ·\sqrt{2} · {r_a}^2 \) \(\displaystyle \ \ ≈{r_a}^2 *2.8284 \)
Long diagonal (\(\small{d}\))
\(\displaystyle d = 2 · r_a \)
\(\displaystyle d = \sqrt{4+2·\sqrt{ 2}} · a \) \(\displaystyle \ \ ≈ 2.613 · a \)
Diagonal (\(\small{e}\))
\(\displaystyle e = 2 · r_i \)
\(\displaystyle e = (1+\sqrt{ 2}) · a \) \(\displaystyle \ \ ≈ 2.414 · a \)
Short diagonal (\(\small{f}\))
\(\displaystyle f = \sqrt{2} · r_a \)
\(\displaystyle f = \sqrt{2+\sqrt{ 2}} · a \) \(\displaystyle \ \ ≈ 1.848 · a \)
Inner circle radius (\(r_i\))
\(\displaystyle r_i=\frac{ 1+\sqrt{2} }{2} ·a\) \(\displaystyle \ \ ≈ 1.207 · a \)
Circumcirle radius (\(r_c\))
\(\displaystyle r_c = \frac{1}{2} · \sqrt{4+2·\sqrt{2}} ·a \) \(\displaystyle \ \ ≈ 1.307 ·a \)
More polygons
Triangle • Square • Pentagon • Hexagon • Heptagon • Octagon • Nonagon • Decagon • Hendecagon • Dodecagon • Hexadecagon • N-Gon • Polygon ring • Concave hexagon • Axial Symmetric Pentagon • Irregular, stretched Hexagon • Irregular, stretched Octagon • Pentagram • Hexagram • Octagram • Star of Lakshmi
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