Nonagon Area Calculator

This regular nonagon area calculator is not only capable of determining the area but also much more. Whether you have the side length, perimeter, distance from center to a vertex, circumscribed circle radius or area, this tool will calculate the remaining values for you. Enter one value and select its unit, then the other values are calculated automatically including the area as main result.

What is a nine belt?

A nonagon is a closed, flat geometric shape with nine straight sides and nine corners. The name comes from the Greek word for "nine" and the suffix "-gon," which means corner or angle. The same figure is also called an enneagon, but that term derives only from its Greek origin. When all nine sides are equal in length and all nine angles are equal in size, it is called a regular nonagon. This tool calculates the area of such regular nonagons.

Nonagons with sides and angles that are not equal are called irregular nonagons. A nonagon is convex when every corner points outward, and concave when at least one corner is pushed inward. Regular nonagons are both convex and fully symmetrical, so their shape can be determined by the length of a single side.

Interesting facts about Nine Ties:

All regular enneagons have the same angles and number of sides no matter what their size. These values can be directly derived from the nine equal length sides. Unlike many other polygons, the angles of an enneagon are whole numbers. This is because 1260 is divisible by nine.

Formula for area of a nonagon

The area of a regular nonagon is determined only by the length of its sides. The standard formula multiplies the square of the side length by a constant factor, which can be expressed as the cotangent of one ninth of a half turn.

Here A is the area and s is the length of one side. Since the cotangent of 20 degrees is approximately 2.7475, the entire factor by which to multiply the square of the side length is about 6.182. Put simply, the area of a regular nonagon is roughly six times the square of its side length.

If the length of a side is unknown but instead the apothem or circumradius are known, then the area can still be calculated. The apothem is the distance from the center to the midpoint of one of the sides, while the circumradius is the distance from the center to one of the vertices.

Here's how to use this calculator:

Enter one of your existing measurements into the appropriate input field. This can be the side length, perimeter, distance from center to a vertex, circumradius or area. Then specify the unit (e.g. centimeters or inches). Once you enter a value, the tool will automatically fill in all other measurements and display the area as the main result. It is not necessary to specify which type of measurement you entered.

If you want to repeat the calculation with a different measurement, first delete the entered values and then enter the new values. By activating the "Comparison" option, you can see the circle that surrounds the nonagon as well as the circle inscribed in the nonagon. By activating the "Diagonals" option, you can check the length of three different diagonals. When opening the diagram, you can see how the area changes when the side length increases.

All measurement values of the nonagon in one place.

Every measurement of a regular nonagon is related to the side length by other measurements. The following table summarizes the relationships used by the calculator so that when you enter one value, the remaining values will be calculated for you.

These lengths are not simple integers or golden ratios like the pentagon or decagon. The ratio of the nonagon stubbornly maintains its irrational properties, which is related to a surprising fact that will be explained later. Although the angles are integers, it is impossible to construct a regular nonagon using only compass and straightedge.

Example calculation:

So let's say we have a regular nonagon with side length of 10 centimeters. If we square the side length, we get 100, and then multiply that by the constant for the nonagon.

So the area of a nonagon with side length 10 centimeters is approximately 618.18 square centimeters. The perimeter of that same nonagon would be 90 centimeters, the distance from its center to any side would be about 13.74 centimeters and the radius of the circumscribed circle would be around 14.62 centimeters. With this calculator you can directly read all these values by simply entering the value for the side length in the corresponding field (10).

Area corresponding to a typical side length

The following table shows some area values corresponding to typical side lengths, rounded to two decimal places. Note that the area increases very quickly as the length of a side becomes larger, since it is squared.

Method of distance from center to side - divided into nine triangles

There is another method of calculating the area which can be used for any regular polygon. Drawing lines from the center to each vertex completely divides the nonagon into nine identical triangles. Since the base of each triangle is equal to one side and the height is equal to the distance from the center to a side, the area is equal to the perimeter multiplied by the distance from the center to a side, divided by two.

This explains why the distance from center to side is so useful. If you can measure the distance from center to center of a side, then you can calculate area without squaring the length of a side. For this reason, this calculator accepts the distance from center to side as an input value.

A nonagon (nine-sided polygon) and its circumscribed and inscribed circles.

Any regular nonagon is inscribed between two circles. The larger circle passes through the nine vertices and its radius equals that of the circumcircle. The smaller circle touches each side at its midpoint, and its radius equals the distance from a side's midpoint to the centroid.

As the shape is constant, the area of a nonagon as a proportion of the larger circle will always be about 92.1%. Compared to the smaller circle, it increases by about 4.3%. This ratio does not change with size. A nonagon has nine sides and appears more circular than pentagons or hexagons because of its proximity to the circumscribing circle. You can show the areas of both circles that make up your nonagon by activating the "Compare" option.

Diagonals of Nine Gacks:

A diagonal is a line segment that connects two non-adjacent vertices. A nonagon has 27 diagonals, which can be verified by 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 27 diagonals actually have only three different lengths. The short diagonals skip one vertex, the medium diagonals skip two vertices, and the long diagonals skip three vertices. Because the nine vertices are evenly spaced, unlike figures with an even number of sides there is no diagonal that passes exactly through the center. The longest diagonal corresponds to the width between opposite vertices, as shown by this calculator.

A nonagon is a geometric object that has some interesting properties. The longer diagonal is exactly equal to the sum of one side length and the shorter diagonal, and this identity holds for all regular nonagons.

The middle diagonal also has special properties. Three vertices, each connected by omitting two vertices, form an equilateral triangle within the nonagon whose side length is equal to the circumradius multiplied by the square root of three. By activating the "Diagonals" option you can display the three different lengths for your nonagon.

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

This is the most surprising secret of the nonagon. Although all its angles are whole numbers, a regular nonagon cannot be constructed exactly with just a compass and straightedge. The reason has to do with one of geometry's oldest problems.

To construct a regular nonagon, one must be able to construct an angle of 40 degrees. An angle of 120 degrees is easy to draw, so trisecting it gives the required 40-degree angle. However, it has been proven that trisecting an angle with compass and straightedge is impossible since 1837, and this is the same obstacle that prevents constructing a nonagon. Algebraically, cosine of 40 degrees is a root of a cubic equation, but this equation does not have any constructible solutions.

In practice one can make a 40 degree angle with a protractor individually and repeat nine times to draw a regular nonagon or use approximation methods. This calculation tool completely avoids this problem by directly using accurate trigonometric functions.

Where do Nine-Necks appear?

Nonagons are not as common as triangles or squares but they can be found in some notable places. For example, all Baha'i Houses of Worship are designed to be nonagons with nine entrances, the number nine standing for unity. The most famous example is the Lotus Temple in New Delhi. Architects looking for a round shape but wanting something special often use nonagons for towers, pavilions and monuments.

Besides architecture, the enneagon can be found on signs and decorative paving stones, as well as in the design of coins and medals, board games and puzzles. It is also a popular example in geometry because it combines both fixed angles and an irrational side length.

Tips for accurate results:

Before using the side length, make sure that you are actually dealing with a regular nonagon. The formula requires all nine sides and angles to be equal. Be consistent with units. If the side length is in centimeters, then the area will be calculated as square centimeters.

Remember that small errors in the length of a side can lead to large errors in calculating area since area is proportional to the square of the length of a side. If possible measure and compare the distance from the center to a side or the radius of the circumscribed circle, and enter those values instead to make sure you are dealing with the same nonagon.