Area Of Octagon Calculator

What this calculator does

A regular octagon is the eight-sided shape with every side the same length and every corner the same angle, the shape of a stop sign or a floor tile. Because it is so even, a single number settles its whole size: the length of one side.

Type the side, pick a unit, and the tool gives you the area inside the octagon.

Using the calculator

1. Type the side length.
2. Pick its unit.
3. Press Calculate.

There is one box, since all eight sides of a regular octagon are equal. The side has to be positive, and the area comes back in square units of the side's unit.

The formula | area = 2 × (1 + √2) × side²

The area of a regular octagon is:

area = 2 × (1 + √2) × side²

Square the side, then multiply by 2 × (1 + √2). That front constant works out to about 4.83, so a quick way to hold it in your head is that a regular octagon covers a little under five times its side squared.

Where the formula comes from

The neatest way to picture it is to start with a square and slice the four corners off at 45 degrees. What is left is a regular octagon. If you work out the big square's area and subtract the four little corner triangles, the algebra tidies up to exactly 2 × (1 + √2) × side².

There is a second route to the same place: split the octagon into eight identical triangles meeting at its centre, work out one, and multiply by eight. Either way you land on the same constant, and the √2 in it traces back to those 45 degree corners.

This is for a regular octagon

One thing to be sure of before you trust the answer: this formula is for a regular octagon, every side the same length and every corner the same 135 degrees, like a stop sign. That is the octagon almost everyone means.

An irregular eight-sided shape, with sides of different lengths, has no single formula. For one of those you would split it into triangles, work out each, and add them up. So check that your octagon is the regular kind, and this one number is all you need.

Units, precision, and rounding

The area comes out in square units of the side's unit, since it is built from a length times a length. The √2 in the formula is an irrational number, its digits running on forever like π, so a regular octagon with a whole-number side never has a perfectly round area. The tool uses a high-precision value of √2 and rounds a long decimal to three places.

A worked example | a 5 cm side

Say the side is 5 cm.

1. Square it: 5² = 25.
2. Multiply by 2 × (1 + √2), about 4.8284: 25 × 4.8284 = 120.711.

So the area is about 120.711 cm². For a real one, a standard US stop sign has sides of about 12.5 inches, which works out to roughly 754 square inches of metal.

Questions people ask

What is the area of a regular octagon with a 5 cm side?

About 120.71 cm². Square the side and multiply by 2 × (1 + √2): 25 × 4.8284.

Does this work for an irregular octagon?

No. This formula is only for a regular octagon, with all sides and angles equal. For an irregular one, split it into triangles, find each area, and add them up.

What is the constant 2 × (1 + √2)?

It is about 4.83, the number you multiply the side squared by. It comes from the octagon's geometry, the same shape you get by cutting the corners off a square.

Is a stop sign a regular octagon?

Yes. A US stop sign is a regular octagon with eight equal sides, each interior angle 135 degrees.

Why is the answer in square units?

Because area is a length times a length. A side in centimetres gives an area in square centimetres.

References

A note on where this comes from. The area of any regular polygon can be found as half its perimeter times its apothem, the distance from the centre to the middle of a side, and for the eight-sided case that standard result simplifies to 2 × (1 + √2) × side². The √2 it contains is irrational, a fact known since the ancient Greeks and traditionally credited to the Pythagorean school. For further reading, see Octagon.

1. The area of a regular polygon as half the perimeter times the apothem, the classical result from which the octagon formula follows.
2. The irrationality of √2, known to the Pythagoreans in antiquity, which is why a regular octagon's area is never an exact round number for a whole-number side.