Nonagon Perimeter Calculator

This regular nonagon perimeter calculator calculates the perimeter of a regular nonagon given its side length, circumradius or incircle radius. But it also can compute anything else about the regular nonagon - just choose what you are given (side, area, circumference, radius of inscribed circle, radius of circumscribed circle) and enter the given value. The other values will be calculated immediately. So if you want to calculate the perimeter of a regular nonagon, this calculator does it for you as the main result. A nonagon is also called an enneagon. This name comes from Greek word meaning "nine" - both names refer to the same geometric object.

What is the perimeter of a nonagon?

A nonagon is a polygon with nine sides and nine corners. A regular nonagon has all sides the same length and all angles the same measure. This tool calculates the properties of a regular nonagon. The perimeter of a polygon (or any other closed curve, such as the circumference of a circle) is the distance around the shape, that is, the total length of its boundaries.

Since a regular nonagon has sides that are all the same length, its perimeter is equal to nine times the length of any one side. You can use whatever unit you wish when measuring, whether it be centimeters or inches.

Here's how to use this tool:

Enter the available measurement into its corresponding field and select the appropriate unit (e.g., centimeters or inches). Once you enter a value, the tool will automatically fill in the other four fields and display the circumference as the main result. It is not necessary to specify which type of measurement you entered first.

If you want to repeat the calculation with a different known value, first delete the previous values entered and then enter the new ones. By opening the compare option, you can compare the nonagon and its circumscribed or inscribed circle. If you open the graph, you can see how the perimeter increases as the side length increases.

Formula for the perimeter of a nonagon

The perimeter of a regular nonagon is determined only by its side length. Since there are nine sides of equal length, simply multiply the side length by nine.

In this formula p represents the perimeter and a represents one side length. Let's say that each side has a length of 10 centimeters. Multiply 10 times 9 to get the perimeter.

If you know the perimeter and want to find the length of a side, just divide by 9.

This is the same rule that applies to any regular polygon, except that nine is replaced by the number of sides. This simple formula only works if all sides are equal in length.

Determine values of in-circle radius, circum-circle radius and area.

It is not strictly necessary to start with the side length. The inradius is the distance from the center to a side, the circumradius is the distance from the center to a vertex, and the area is the size of the interior. All these quantities are connected by geometric relationships involving the nine triangles that make up the nonagon, so if one value is known, then the side length can be calculated and hence the perimeter.

In these formulas r is the radius of the inscribed circle, R is the radius of the circumscribed circle and A is the area. This angle is the result of dividing 180 by 9, which is exactly 20 degrees. However, the resulting constants are irrational numbers. Since calculations of trigonometric functions are performed by a calculator, you only need to enter the available values. Unlike pentagons or decagons, these nonagon constants do not hide simple integers or the golden ratio. This is because a nonagon cannot be constructed with just a compass and straightedge.

Check all of the measurements for the nonagon as a summary.

Each measurement of a regular nonagon is related to any other measurement by the side length. The following table summarizes the relationships used by the calculator so that when one value is entered, the others can be calculated.

Example calculation

The following table gives some examples of the perimeter that is associated with common side lengths. The perimeter increases as the length of the sides increase and they are both directly proportional to each other since the length of the sides is not squared.

It is just as easy to derive the perimeter from the side length. For a regular nonagon with a circumference of 108 centimeters, the side length is the value that results from dividing 108 by 9, i.e. a side length of 12 centimeters. If you enter 108 in the field for the perimeter, the side length will be displayed immediately, as well as the distance from the center to a corner, the radius of the circumcircle and the area.

Three types of diagonals in a nonagon

A diagonal is a line segment that connects two non-adjacent vertices. A nonagon has 27 diagonals, which can be verified using the formula for polygons: multiply the number of sides by a number three less than this and then divide the result by two.

These diagonals have three different lengths because the distance between vertices is either two, three or four. In the calculator they are called short diagonal, medium diagonal and long diagonal respectively. There are two simple relationships between them: the long diagonal is exactly one side length longer than the short diagonal, which is a special property of the nonagon. The medium diagonal is equal to the square root of three times the circumradius. This is because nine medium diagonals form the sides of three overlapping equilateral triangles, and nine is the product of three and three.

Angle of a regular nonagon

The angles of a regular nonagon are constant and always result in nice whole numbers regardless of the size it is drawn. Each of the nine interior angles measures exactly 140 degrees, and their sum is 1260 degrees. The central angles formed by lines drawn from the center to two adjacent vertices also measure exactly 40 degrees, as do the exterior angles.

Why can't a regular nonagon be constructed with compass and ruler?

Constructing a regular nonagon corresponds to cutting off a 40 degree angle from a circle, which means trisecting a 60-degree angle, and with just these two tools it is not possible to generally trisect angles.

This also explains why the constants describing the relationship between side length and distance from center as well as between side length and circumradius do not have a simple form of a square root, like for example in the case of the corresponding constants for a pentagon. The constants for the pentagon can be expressed by the golden ratio, while the constants for the nonagon are values of trigonometric functions which are irrational numbers. Although these values cannot be produced with classical construction methods, they do not affect the possibility to use them.

Regular and irregular nonagons

This calculator assumes that a nonagon is a regular nonagon, meaning that all nine sides are the same length. If a nonagon is irregular and each side has a different length, then the simple formula of multiplying the sum of the side lengths by nine cannot be used. In this case, the perimeter can still be found by adding up the lengths of all nine sides. The formulas for the distance from center, circumradius, and area only apply to regular nonagons.

The purpose of the perimeter of a nonagon.

The nonagon is used in design, architecture and handicrafts. In Bahá'í Houses of Worship the floor plan is based on a nonagon with an entrance assigned to each direction, giving it actual symbolic meaning. Nonagons are also used for decorative tiles, flower beds, gazebos, coins, mirrors and signs.

Knowing the circumference allows a manufacturer to determine how much material (e.g. siding, profiles, fence, frames) is required to cover the entire perimeter of the nongon. Designers and draftsmen use the same relationship to fit a nongon into a given circle or to determine the size of a circle that will just enclose a nongon.

Tips for accurate results:

Carefully measure each side and use a calculator to multiply rather than attempting to calculate the entire perimeter at once. If the lengths of the sides are given in centimeters, then your perimeter will be expressed in centimeters as well, so units will be consistent. In the formula for a regular nonagon, it is important that all nine sides are the same length. Therefore, before proceeding with calculations using the apothem, circumradius or area, verify that you have a true regular nonagon.