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Properties of a N-gon
Properties of a Hexagon
Properties of a Decagon
Properties of a Rectangle
Properties of a Circular area
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Properties of an Octagon

- By Dr. Minas E. Lemonis, PhD - Updated: March 3, 2019

This tool calculates the basic geometric properties of a regular octagon. Regular polygons are equilateral (all sides equal). The tool can calculate the properties of the octagon, given the length of its sides, or the inradius or the circumradius. Enter below the shape dimensions. The calculated results will have the same units as your input. Please use consistent units for any input.

Known data:
Geometric properties:
Area =
Perimeter =
Side α =
Inradius Ri =
Circumradius Rc =
Angles :
Interior φ =
Central θ =
shape details
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Definitions

Geometry

Octagon is a polygon with eight sides and eight vertices. An octagon is regular when all its sides are equal. The interior angles of a regular octagon are always 135°. For any regular polygon, a circle that passes through all vertices can be drawn. That is the cirmuscribed circle or circumcircle of the polygon. Also, a circle that is tangent to all sides can be drawn, which is called inscribed circle or incircle.

The radius of circumcircle Rc and incircle Ri (usually called circumradius and inradius respectively), are related to the side length α and also among each other. These relationships can be discovered using the properties of the right triangle, highlighted in the figure below, leading to the following formulas:

\[ \begin{split} R_c & = \frac{a}{2 \sin{\frac{\theta}{2}}} \\ R_i & = \frac{a}{2 \tan{\frac{\theta}{2}}} \\ R_i & = R_c \cos{\frac{\theta}{2}} \end{split} \]

where θ the central angle and α the side length. For the regular octagon, θ = 45°, therefore the above formulas are simplified to:

\[ \begin{split} R_c & = \frac{a}{2 \sin{22.5^{\circ}}} \approx 1.31 a \\ R_i & = \frac{a}{2 \tan{22.5^{\circ}}} \approx 1.21 a \\ \\ R_i & = R_c \cos{22.5^{\circ}} \approx 0.92 R_c \end{split} \]

shape geometry

The area of any regular N-sided polygon can be found, considering that the whole shape can be decomposed to N number of identical isosceles triangles (sides α,Rc ,Rc ). The area of each triangle is \(\frac{1}{2}a R_i\). Therefore for the polygon:

\[ A = \frac{N}{2} a R_i = \frac{N a^2}{4\tan{\frac{\theta}{2}}} \]

Substituting for the specific case of regular octagon, the above formula becomes:

\[ A = \frac{5 a^2}{4 \tan{22.5^{\circ}}} \approx 4.83 a^2 \]

The perimeter of any N-sided regular polygon is simply the sum of the lengths of all sides: \( P = N a \). Therefore, for the regular octagon :

\[ P = 8a \]

NOTE: Some of the above formulas have general scope for all regular polygons. These are the formulas, with θ, φ or N variables, not substituted with specific values.

See also
Properties of a N-gon
Properties of a Hexagon
Properties of a Decagon
Properties of a Rectangle
Properties of a Circular area
All Geometric Shapes tools