Octagon Perimeter Calculator

This octagon perimeter calculator allows you to find the perimeter of a regular octagon which is an eight-sided polygon with all sides equal length and angles equal measure. However, it also provides more functionality such as being able to compute the other dimensions of the shape when given any one dimension. You only need to input one of the available measurements (side length, perimeter, distance from center to side, circumradius or area) and all others will be calculated automatically. By entering a value for one measurement and selecting its unit, the other values are filled in automatically with the perimeter being shown as the main result. The name octagon comes from the Greek word "okto" meaning eight. A common example of an octagon is probably that red parking sign on the side of the road.

What is the perimeter of an octagon?

An octagon is an 8-sided polygon with eight angles. A regular octagon has sides of equal length and all interior angles are the same size. This tool will calculate the perimeter of a regular octagon. The perimeter is the total distance around the outside, or the boundary, of a shape.

Since a regular octagon has all sides of equal length, the perimeter is eight times the length of one side. It can be measured in common units such as centimeters or inches. The perimeter is the only measurement for an octagon that does not require the use of square roots to calculate. It can be easily determined by multiplying 8 by the length of a single side. In contrast, the distance from the center to a side, the radius of the circumscribed circle, and the area all involve nested roots specific to the octagon.

How to use this calculator:

Enter a given measurement into the appropriate field and select its unit (e.g., centimeters or inches). After entering one value, the tool will automatically populate the remaining four fields and display the circumference as the main result. It is not necessary to specify which measurement was originally entered.

If you want to redo the calculation with a different known value, first delete the previous values entered and then enter the new ones. Besides the answer, it will also show the distance between opposite sides, the distance between opposite vertices, and the short diagonal. By opening the compare option, you can compare an octagon with its circumscribed circle and inscribed circle. By opening the graph, you can see how the perimeter changes as the side length increases.

Formula for the perimeter of an octagon

The perimeter of a regular octagon is determined by the length of one side multiplied by eight since an octagon is composed of eight equal sides.

In this formula, P is the perimeter and a is the length of one side. If each side has a length of 5 cm, you would get the perimeter by multiplying that number by eight.

Conversely, if you know the perimeter and want to find the length of one side, simply divide by 8.

This rule applies to any regular polygon, you just have to replace the eight in the formula with the number of sides that the polygon has. However, this simple formula only works for polygons whose sides are all equal length.

Calculation based on side center distance, circumcircle radius or area.

It is not always possible to start with the side length. The apothem is the distance from the center to a midpoint of one of the sides, the circumradius is the distance from the center to one of the vertices and the area describes the space inside the figure. These quantities are all geometrically related to each other and to the side length since they are determined by the relationships of the eight triangles that make up the octagon. Therefore, it is possible to calculate the side length backwards if any of these values are known, from which the perimeter can be derived.

In these formulas r is the inradius, R is the circumradius and A is the area. This angle is exactly 22.5 degrees which is what you get when you divide 180 by 8. Unlike heptagons or nonagons an octagon is a polygon that can be constructed exactly with compass and straightedge. So the constants in the formulas are expressed in simple closed form involving square roots of two. The inradius is exactly equal to what you get when you divide (square root of 2 plus 1) by 2 and then multiply this result by the side length. The area is exactly equal to what you get when you multiply the sum of 1 and the square root of 2 by 2, and then multiply this result by the square of the side length. The trigonometric function calculations are done by a calculator so all you need to do is enter your values.

All measurement values of an octagon

Every measurement of a regular octagon is related to the others by the length of its sides. The following table shows the relationships used in the calculator so that you can enter one value and find the remaining values.

Examples of calculations

The following table shows the perimeter for some common side lengths. Note that as the side length increases, so does the perimeter, and it is directly proportional to the side length since the side length is not squared.

Calculating the side length from the perimeter is also simple. If a regular octagon has a perimeter of 96 centimeters, then the side length is 12 centimeters, which is the result of dividing 96 by 8. Entering 96 in the Perimeter column will immediately display the side length, as well as the distance from the center to a side, the radius of the circumscribed circle, and the area.

Opposite sides and opposite corners distance

Since an octagon has eight sides, there are pairs of opposite sides and pairs of opposite vertices, which gives two natural ways to measure a width. The distance between opposite sides is the distance between any two opposite sides, equal to twice the apothem. The distance between opposite vertices is the distance between any two opposite vertices, equal to twice the circumradius, or longest diagonal in the octagon.

Regular octagons have a special property in this regard. Since eight is a multiple of four, there are simultaneously flat sides pointing up, down, left and right, so that the distance between opposite sides is equal regardless of direction. The tightest frame that can enclose an octagon is a square, with the distance between opposite vertices along the diagonal of that square. If you have ever used a wrench to tighten a nut on an octagon, then the distance between opposite sides is the size of the wrench you use.

The silver ratio in an octagon.

The pentagon and decagon are famous for containing the golden ratio. The regular octagon contains a closely related proportion known as the silver ratio. The silver ratio is one plus the square root of two, approximately 2.4142. This ratio results simply from comparing the distance between opposite sides to the length of a side in an octagon.

The same silver ratio is also the value obtained by dividing the mean diagonal of an octagon by its side length, or by doubling the distance from center to side and then dividing it by the side length. Thus the distance between opposite sides, the mean diagonal, and double the distance from center to side are all equal in length and are equal to the product of the silver ratio and the side length. Like the golden ratio for the pentagon, this is a number that represents the octagon.

Diagonals of an octagon:

A diagonal is a line that connects two vertices that are not directly next to each other. An octagon has 20 diagonals, which can be verified by the formula for polygons: multiply the number of sides by a number that is three less than this, and then divide the result by two.

These diagonals have three different lengths. This is because from a vertex there can be two, three or four vertices away to another vertex. The short diagonal connects two vertices that are two apart. The medium diagonal connects three vertices that are three apart and is equal in length to the distance between opposite sides. The long diagonal connects four vertices that are four apart and passes through the center point. This is equal in length to the distance between opposite vertices, which is twice the radius of the circumscribed circle. Drawing all the medium diagonals creates a contour of an eight-pointed star, while drawing the short diagonals creates the outline of the Star of Lakshmi. In the center there is another regular octagon.

Angle of a regular octagon:

The angles of a regular octagon are fixed and no matter how large or small it is drawn the result will always be an integer. Each of its eight interior angles measures exactly 135 degrees and their sum is 1080 degrees. The central angle, formed by two adjacent vertices through the center point, also measures 45 degrees as do the exterior angles.

A regular octagon can be constructed using a compass and straightedge.

A regular octagon is one of the polygons that can be constructed exactly with a compass and straightedge. The quickest method is to start with a square and cut off congruent sections from each corner so as to increase the number of sides from four to eight. Another way is to first draw a circle, then mark four points on it which are equally spaced around the circumference, and then bisect each of the four quarter-circle arcs to find the remaining four vertices.

Thus the constants between side length and distance to center as well as between side length and circumradius can be expressed as simple root terms consisting of square roots of two. No cumbersome trigonometric values like in heptagon or nonagon are required. As eight is a power of two, the regular octagon meets the requirement that a regular polygon can be repeatedly halved to construct it.

Regular octagon and irregular octagon

This tool assumes that you have a regular octagon, which means all eight sides are the same length. If your octagon is irregular and has different side lengths, then the simple formula of multiplying the length by 8 won't work. In this case, the perimeter will still be the sum of the lengths of each side, so you'll need to add them individually. The formulas for the distance from center, circumradius, and area only apply to regular octagons.

The use of the perimeter of an octagon

The shape of the octagon is ubiquitous in our everyday life. Park signs are a globally known example of an octagon as this shape is used to make them recognizable by their outline even without reading the inscription. Also, nuts, screw heads, pavilions, villas, umbrellas, plates, mirrors and tiles use the same shape, and cages for mixed martial arts fights are called "octagons".

The octagon has also had a long symbolic meaning. The Dome of the Rock in Jerusalem was built on an octagonal floor plan and the Tower of the Winds in Athens is a famous example of the oldest known architecture with an octagonal ground plan, built in the first century BCE. For manufacturers, the perimeter gives insight into how many quantities of materials are needed to cover the outline of an octagon, such as cladding, decoration, fencing or frames. Architects use the same relationship to fit an octagon inside a given circle.

Tips for accurate results

Measure each side carefully and do not try to measure the entire perimeter at once. Leave the multiplication to a calculator. If you are measuring in centimeters make sure that your perimeter is also in centimeters so that your units are consistent. The formula for a regular octagon requires all eight sides be of equal length. So before calculating the perimeter from either the apothem, circumradius or area, verify that your shape is actually a regular octagon.

This tool is for general educational purposes and everyday planning. For construction work, engineering or other risky activities these values should be checked against the required standards and tolerances of your project.