Decagon Area Calculator

This regular decagon area calculator is not only able to calculate the area of a regular decagon but also has many other uses. If you input one known measurement, either side length, perimeter, inradius, circumradius or even the area itself, all other values can be calculated.

By entering a value and selecting the unit it will automatically display the remaining results with the area being highlighted as the most important result.

What is a decagon?

A decagon is a closed, plane figure with ten straight sides and ten angles. The name comes from the Greek word for "ten corners". When all ten sides are equal in length and all ten angles are equal in size it is called a regular decagon, which is exactly what this tool deals with.

Decagons that do not have equal sides and angles are called irregular decagons. A convex decagon has all of its vertices pointing outwards while a concave-convex decagon has at least one vertex pointing inwards. Since a regular decagon is a convex shape with full symmetry, it can be fully described by giving just the length of a single side.

Data overview for decagons:

Regardless of size, all regular decagons have the same ratio between angles and sizes. These values can be directly derived from the ten equal sides.

Formula for area of a decagon

The area of a regular decagon is determined only by its side length. The standard formula multiplies the square of the side length by a constant number.

In this formula A represents the area and s represents the side length. Since the value under the square root is approximately 3.0777, the entire factor that is multiplied by the square of the side length is approximately 7.694. In other words, the area of a regular decagon is about equal to the product of the side length squared and 7.694.

Even if the side length is unknown, the area can be calculated if the apothem or circumradius are known. The apothem is the distance from the center to the midpoint of a side, while the circumradius is the distance from the center to one of the vertices.

How to use this calculator:

Enter one of the known values (side length, perimeter, distance from center to side, circumscribed circle radius or area) into its corresponding field and select a unit such as 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 of the entered values is which.

If you want to pick another size and redo the calculations, just delete the previous fields first then enter new values. By opening the compare option you can see the outer circle surrounding the decagon as well as the inner circle inside of the decagon. When you open the graph you will be able to observe how the area changes when the length of sides increases.

All measurement values of the decagon

Any measurement of a regular decagon is connected to other measurements by the side length. The following table shows the relationships that the tool uses so you can enter any one value and calculate the rest.

The radius of the outer circle has a special property; in a regular decagon, it is exactly equal to the side length multiplied by the golden ratio, which is approximately 1.618. Thus there is the same relationship between the circumcircle and this number as for pentagons or pentagrams.

A complete example

Suppose there is a regular decagon with side length of 10 centimeters. Square the side length to get 100, then multiply this by the decagon constant.

So the area of a decagon with side length 10 centimeters is approximately 769.42 square centimeters. The perimeter of that same decagon would be 100 centimeters, the apothem would be about 15.39 centimeters and the radius of the circumscribed circle would be about 16.18 centimeters. You can use this calculator to find all those results directly by simply entering a value for the side length into the corresponding field (10).

Area corresponding to a typical side length

The following table shows some area values corresponding to typical side lengths. The results are rounded to two decimal places. Note that the area increases rapidly as it is proportional to the square of the side length.

Method of distance from center to side: another representation with ten triangles

There is another way to understand the area, and this applies for all regular polygons. Drawing a line from the center to each vertex divides the decagon into exactly ten identical triangles. Since the base of each triangle is equal to the side length, and the height is equal to the distance from the center to the side, the area comes out to be the perimeter multiplied by the distance from the center to the side, divided by two.

This is 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 the area without having to square the length of a side. The slide rule accepts the distance from center to side as an input for exactly this reason.

A decagon with its inscribed and circumscribed circles.

All regular decagons will fit neatly between two circles. The larger circle passes through the ten vertices, and its radius is the circumradius. The smaller circle touches all sides at their midpoints, and its radius is the apothem.

Because the shape of the figure is fixed, its area will always be about 93.5 percent of the area of the larger circle and about 3.4 percent more than the area of the inscribed circle. These ratios remain constant regardless of size. This shows just how close a decagon comes to being an actual circle. If you open up the compare options above, you can find out what areas of both circles correspond with your decagon.

Diagonals of a decagon.

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

These diagonals have three different lengths. This is because the vertices can be separated by two, three or four other vertices. The longest diagonal runs from one vertex directly through the center to the opposite vertex. This longest diagonal is the width between the vertices that the calculator outputs.

Use of decagons

Decagons are more common than you might think. Some coins are minted in the shape of a decagon to make them tactilely distinguishable. Architects sometimes use decagonal floor plans for towers, pavilions and baptisteries.

Designers incorporate decagons in logos, watch cases, mirrors and tile patterns, while jewelers use decagonal facets. In classrooms, the regular decagon is a frequent example that connects a familiar pentagon with a smooth circular shape.

Tips for accurate results

Before you rely on the length of just one side, make sure that it is indeed a regular decagon. The formula assumes that all ten sides and all ten angles are equal. Always be careful to match units. A side length measured in centimeters will give an area in square centimeters.

Since area varies as the square of side length, a small error in measuring the sides will result in a large error in the area. If possible, also measure and compare the distance from the center to any side (or radius of the circumscribed circle), enter those values here, and make sure you get the same decagon area result.