About

Geometric precision is often required in architecture, tiling, and machining, where the octagon is a frequently used shape. Whether you are cutting a tabletop from a circular slab or designing a stop sign, understanding the relationship between the octagon and its bounding circle is crucial.

This Inscribed Octagon Calculator determines all key properties of a regular octagon based on a single known variable. By defining the circumradius (the radius of the circle containing the octagon), the tool instantly computes the side length, the apothem (inradius), and the total area. It includes a dynamic visualizer to help students and professionals verify their parameters visually.

Formulas

For a regular octagon inscribed in a circle of radius R, the Side Length (a) is calculated using trigonometry:

a = 2R sin(22.5°) ≈ 0.765R

The Area (A) can be found using the side length:

A = 2(1 + √2) a²

The Apothem (r), or distance from center to midpoint of a side:

r = R cos(22.5°)

Reference Data

Parameter	Formula (in terms of Radius R)	Value for R=1	Value for R=10
Side Length (a)	R × 0.76537	0.765	7.654
Apothem (r)	R × 0.92388	0.924	9.239
Area (A)	2.828 × R²	2.828	282.843
Perimeter (P)	8 × a	6.123	61.229
Interior Angle	135°	135°	135°
Exterior Angle	45°	45°	45°

Frequently Asked Questions

An inscribed octagon is a regular eight-sided polygon drawn inside a circle such that all eight vertices (corners) touch the circle's circumference. The radius of this circle is called the circumradius.

If by "width" you mean the total span from one flat side to the opposite flat side (the diameter of the incircle), then the side length a = Width × (√2 - 1) ≈ Width × 0.414.

No. The radius (circumradius) is the distance from the center to a corner. The apothem (inradius) is the shorter distance from the center to the midpoint of a flat side.

No, this tool assumes a regular octagon where all sides and angles are equal. Irregular octagons require coordinate geometry or specific side lengths/angles to solve.