Octagon calculator

What is a regular octagon?

An octagon is a polygon with eight sides and eight vertices. In this calculator, we work with the regular octagon: all eight sides have the same length and all eight interior angles are equal. Once one length is known, the entire shape is determined by geometry.

That distinction matters. A general octagon can have unequal sides and unequal angles, so one side length is not enough to determine its area. A regular octagon is different because symmetry fixes the relationships between side, area, perimeter, diagonals, circumradius and inradius.

Every important octagon formula, explained

Let a be the side length. The formulas below are not isolated facts: they are connected through the same 45° central geometry.

Why does the area formula work?

Draw lines from the center to all eight vertices. The octagon becomes 8 congruent isosceles triangles. Each triangle has base a and height r, so its area is a·r/2. Multiplying by eight gives:

For a regular octagon, r = a(1 + √2)/2. Substituting that into 4ar gives A = 2(1 + √2)a². This is useful because it explains the formula instead of asking you to memorize it.

Useful inverse formulas

Worked examples, step by step

Example 1 — side length 5 cm

Suppose the side is a = 5 cm. Start with the perimeter and area, then use the diagonal and radius relationships.

Example 2 — area is known

Suppose a drawing gives A = 200 cm², but the side is missing. Rearrange the area formula rather than guessing:

Now that a is known, the calculator can derive the perimeter, three diagonal classes and both radii from that single recovered side length.

Example 3 — perimeter is known

If P = 40 cm, the quickest route is a = P/8 = 5 cm. Notice how this immediately returns the same values as Example 1. This is a useful self-check when working from a technical drawing.

Where do 45°, 22.5° and √2 come from?

A full turn around the center is 360°. Because a regular octagon has eight identical sectors, each central angle is 360°/8 = 45°. Draw the radius to the midpoint of one side and the 45° triangle is split into two right triangles, each with an angle of 22.5°.

Using right-triangle trigonometry gives the general regular-polygon relationships r = a/[2 tan(π/8)] and R = a/[2 sin(π/8)]. For the octagon, those expressions simplify to the familiar formulas containing √2.

A second way to understand the area

Imagine a square surrounding the regular octagon. Four identical right triangles are removed from its corners. This construction is especially useful when drawing or fabricating an octagon from a square, because it explains why the octagon has four pairs of parallel sides and why its corners are cut at equal geometry.

How the 20 diagonals are organized

A polygon with n vertices has n(n − 3)/2 diagonals. For an octagon this gives 8 × 5 / 2 = 20. Those 20 segments are not all different: in a regular octagon they fall into three length classes.

The long diagonal is especially important: it passes through the center and is therefore the diameter of the circumcircle. The medium diagonal corresponds to the distance across opposite parallel sides, so it is twice the apothem.

Circumradius vs. inradius

How octagon measurements are used in practice

Different projects start with different measurements. A tile maker may know the side length, a drawing may specify the distance across flats, and a circular layout may give a radius. Knowing which measurement you have prevents using the wrong formula.

Five mistakes to avoid

1. Confusing any octagon with a regular octagon. The formulas on this page assume all eight sides and angles are equal.
2. Using a² incorrectly. Area uses the square of the side, so the unit becomes cm², m², in², and so on.
3. Calling every diagonal the same. A regular octagon has short, medium and long diagonals.
4. Mixing R and r. R goes to a vertex; r goes perpendicularly to a side.
5. Forgetting the scale effect. If the side doubles, perimeter doubles but area becomes four times as large.

Octagon questions, answered clearly