Octagon Radius Calculator - Calculator Academy

Calculate an octagon's side length or radius from one known value and quickly convert between units in mm, cm, m, km, in, ft, yd, and mi.

Octagon Radius Formula

The octagon radius calculator finds the circumradius of a regular octagon, which is the distance from the center of the shape to any vertex. This value is useful in layout work, fabrication, drafting, geometry, and any design where the octagon must fit inside a circle or be built from a center point.

R = \frac{S}{2\sin\left(\frac{\pi}{8}\right)}

Where:

R = radius of the regular octagon
S = side length of the regular octagon

This can also be written in an equivalent simplified form:

R = \frac{S}{\sqrt{2-\sqrt{2}}}

The radius is always in the same unit as the side length. If the side length is entered in inches, the radius is returned in inches. If the side length is entered in meters, the radius is returned in meters.

How the Calculation Works

A regular octagon has 8 equal sides, so the central angle between adjacent vertices is 45°. If that angle is split in half, a right triangle is formed with a 22.5° angle. The side length acts as a chord of the circumscribed circle, which leads directly to the radius relationship below:

S = 2R\sin\left(\frac{\pi}{8}\right)

Solving that expression for the radius gives the calculator formula shown above.

How to Use the Octagon Radius Calculator

Enter the side length of the regular octagon.
Select the appropriate unit if needed.
Calculate the result to get the radius from the center to a vertex.
Use the result for layout, area calculations, diameter calculations, or construction geometry.

Example

If the side length of a regular octagon is 10, the radius is:

R = \frac{10}{2\sin\left(\frac{\pi}{8}\right)} \approx 13.066

So a regular octagon with side length 10 has a radius of approximately 13.066 units.

Related Regular Octagon Formulas

Once the radius is known, several other important octagon dimensions can be found quickly.

Measurement	Formula	Use
Perimeter	$P = 8S$	Total distance around the octagon
Apothem	$a = R\cos\left(\frac{\pi}{8}\right)$	Distance from center to the midpoint of a side
Apothem from side length	$a = \frac{S}{2\tan\left(\frac{\pi}{8}\right)}$	Useful when radius is not yet known
Area	$A = 2\left(1+\sqrt{2}\right)S^2$	Surface area of the regular octagon
Vertex-to-vertex diameter	$D_v = 2R$	Maximum width across opposite vertices
Flat-to-flat width	$W_f = 2a$	Width across opposite sides

Radius vs. Apothem

These two measurements are often confused:

Radius goes from the center to a vertex.
Apothem goes from the center to the middle of a side.

In a regular octagon, the radius is always longer than the apothem.

When This Formula Applies

This calculator and formula apply to a regular octagon only, meaning:

all 8 sides are equal, and
all interior angles are equal.

If the octagon is irregular, there is not a single universal radius formula because the distance from the center to each vertex may not be the same.

Practical Uses of the Octagon Radius

laying out an octagon from a center point
building octagonal decks, tables, windows, signs, or columns
finding the circle that passes through all 8 vertices
converting between side length, diameter, and area
CAD, machining, woodworking, and architectural drafting

Common Questions

Is the octagon radius the same as half the width?

Not always. Half the vertex-to-vertex width equals the radius, but half the flat-to-flat width equals the apothem instead.

Does changing units affect the formula?

No. The formula stays the same in any unit system. Only the numeric unit label changes.

Can I use decimals for the side length?

Yes. The formula works for whole numbers, decimals, and fractional dimensions as long as the octagon is regular.

Why is the result larger than the side length?

For a regular octagon, the distance from the center to a vertex is greater than the length of one side, so the radius is expected to be larger than the side length.