Octagon Area Calculator

This regular octagon area calculator can calculate the area of a regular octagon, but also calculates other values of interest of such an octagon. You only have to enter one single measure - whether it is the side length, perimeter, short diagonal, medium diagonal, long diagonal, circumradius, inradius or even the area itself - and all other values will be calculated for you automatically.

After you enter a value and select the unit, the remaining values will be automatically displayed with area highlighted as the most important result. As a regular octagon is fully defined by a single measurement, there are no menus to set up beforehand.

What is an octagon?

An octagon is a closed, plane figure that has eight straight sides, eight vertices, and eight interior angles. The name comes from the Greek word for "eight corners." The same root gives us the word octopus, which has eight arms.

If all eight sides are the same length and all eight angles are equal, then the octagon is called a regular octagon. This type of octagon is what we will be dealing with in this tool. Octagons that do not have equal side lengths or angles are called irregular octagons, and their area must be calculated differently.

A regular octagon is a convex polygon, meaning that all of its vertices lie outside the figure. It also has high symmetry and possesses eight axes of symmetry as well as an eighth rotational symmetry. Because of these symmetries, the entire shape can be defined by the length of just one side.

Basic facts about octagons

Regardless of size, all regular octagons have the same properties with regard to angles and number. These values are directly derived from the eight equal length sides.

Formula for calculating area of an octagon

The area of a regular octagon is determined only by the length of its sides, and in a standard formula it's the square of the side length multiplied by a constant number.

Here A is the area and a is the length of one side. As this constant is approximately 4.828, it can be said that the area of a regular octagon is about 4.828 times the square of its edge length.

If the distance from the center to a side or the radius of the circumscribed circle is known, these values can be used to calculate the area. The distance r from the center to the midpoint of a side and the radius R of the circumscribed circle, which represents the distance from the center to a vertex, can be used for this purpose.

The method of calculating the inradius and the eight triangles

To understand where the formula comes from, it is easiest to draw lines from the center to each vertex. This divides the octagon into eight congruent triangles with base equal to one side of the octagon and height equal to the inradius.

If you add these values together, you get the value that is created by multiplying half of the circumference with the inradius. This is a general formula which applies to all regular polygons.

This calculator shows the area of each of the eight triangles on the result map so you can see how your octagon is divided up. If you can measure the inradius - that is, the distance from the center to the midpoint of a side - then you can calculate the area directly. That's why the inradius is an input field.

Remove a square with four corners

There is another particularly illustrative representation for the octagon. If you cut off a right triangle from each of the four corners of a square, what remains is a regular octagon. This is referred to in mathematics as cutting off the corners of a square and it is the easiest method for drawing an octagon by hand.

The side length of the first square is equal to the width between the parallel sides of the octagon, which is the length obtained by multiplying the side length by the square root of 2 plus the side length itself. The four triangles that are cut off at the corners are important. Regardless of the size of the octagon, their sum always results in the square of a side.

If you subtract the square of this side from the area of the square, you get directly to the formula for the octagon. A regular octagon occupies about 82.8 percent of the area of the surrounding square. You can check the areas of your octagon and the square under the "cut out square" option.

All measurement values of the octagon in one place

Each measurement of a regular octagon interacts with the other measurements through the side length. The following table shows the relationships used in the calculation tool so that when you enter one value, it can calculate the remaining values for you.

A complete calculation example

Let's say we have a regular octagon with side length of 10 centimeters. We square the side length to get 100, and then multiply it by the constant for an octagon.

So the area of an octagon with a side length of 10 centimeters is approximately 482.84 square centimeters. The perimeter of that same octagon would be 80 centimeters, the length of its long diagonal would be about 26.13 centimeters, the width between two parallel sides would be approximately 24.14 centimeters, the radius of the circumscribed circle would be around 13.07 centimeters and the distance from the center to a side would be about 12.07 centimeters.

If you use this calculator, all you have to do is enter the value 10 into the field for side length to get these values directly. You can also start with other measurements and then the calculator will show you the side length.

Examples of areas for common side lengths

The following table shows the area for some common side lengths rounded to two decimal places. Note that the area increases quickly as the side length becomes larger since the side length is squared.

Diagonals of an octagon

A diagonal is a line segment that connects two non-adjacent vertices. An octagon has twenty diagonals. This can be verified using the formula for the number of diagonals in a polygon: the number of sides multiplied by (the number of sides minus three), divided by two.

These twenty diagonals have three different lengths. The short diagonals cross one vertex, and their length is the square of the sum of the side length and the square root of two. The medium diagonals cross two vertices, and their length is the product of the side length and the sum of one and the square root of two. This also corresponds to the height of the octagon, i.e. the distance between two opposite sides. The long diagonals run from a vertex through the center point to the opposite vertex, corresponding to the width between two opposite vertices. Their length is exactly twice the radius of the circumcircle.

Octagon and Silver Ratio

Like the regular pentagon, the regular octagon contains the silver ratio, which is one plus the square root of two, approximately 2.414. This is the ratio between the length of a median diagonal and the side length, as well as the ratio between the width between any two parallel sides and the side length.

The silver ratio can also be used to relate the side length with the distance from the center to a side. This distance is half of that. The same relation occurs when cutting off corners of a square to form an octagon. Just as the golden ratio is a number characteristic for the regular pentagon, the silver ratio is a number characterizing the octagon.

Octagon and its circles

Any regular octagon will fit snugly between two circles. The larger circle passes through all eight vertices and has a radius equal to the circumradius of the octagon. The smaller circle touches each side at its midpoint, and has a radius equal to the apothem of the octagon.

As the shape of the figure is constant, its area will always be about 90.0 percent of the larger circle and about 5.5 percent more than that of the smaller one. These proportions do not change regardless of size. An octagon is often used instead of a hexagon because it approximates to the full circle better and has both a circular outline and flat sides which are easy to make.

Uses of octagon

The most common example of an octagon is probably the stop sign. Because road authorities only use octagons for stop signs, drivers can recognize a stop sign by its shape even if it's viewed from behind or obscured by snow. This shape is so distinctive that it's recognizable at a glance.

Octagons are popular in architecture and design because their flat sides are easy to construct and the outline is nearly circular. The famous Octagon House in Washington D.C. was built for Colonel John Tayloe III and got its name from its shape, as does the floor plan. For the same reason octagons are often used on umbrellas, gazebos, and summer houses.

Octagonal tiles are a classic. They can be combined with small squares to create a seamless floor and this pattern is still often found in kitchens and bathrooms today. Octagons can also be found inside cameras. Many lenses have an octagonal aperture, which causes bright light to appear as eight-rayed starbursts in photographs. The cages used in mixed martial arts are also octagonal and are commonly referred to simply as "octagons."

Tips for accurate results

Before proceeding with this method, make sure that your shape is actually a regular octagon. The formula requires that all eight sides and angles are identical to each other. Ensure that the units you use for measurement are consistent throughout. If the length of one side is measured in centimeters, find the area in square centimeters.

Note that area changes in proportion to the square of side length so small errors in side length can lead to large errors in area. If possible measure from center to a side, diagonal or between two parallel sides and check each other's measurements, enter them here and make sure they give the same octagon result.