Heptagon Area Calculator

This heptagon area calculation tool allows you to find the area of a regular heptagon as well as other calculations. Whether given the side length, perimeter, inradius, circumradius or just the area, all other values can be calculated.

By entering one value and selecting its unit, the remaining results will automatically be displayed with the area being highlighted as the most important result.

What is a heptagon?

A heptagon is a closed, plane figure with seven straight sides and seven corners. The name comes from the Greek word for "seven corners," and the same shape can also be called a septagon. This term combines a Latin prefix with a Greek suffix. When all seven sides are equal in length and all seven angles are equal in size, it is a regular heptagon, which is the form this calculator deals with.

Seven-sided polygons in which not all sides and angles are equal are referred to as irregular heptagons. If all vertices point outwards, the figure is a convex heptagon; if at least one vertex points inwards, it is a concave heptagon. Regular heptagons are convex figures with complete symmetry, so they can be fully determined by the length of one side.

Overview of data for a heptagon:

Regardless of size, all regular heptagons have the same ratio between angles and sizes. These values are derived directly from the seven equal sides. The interior angles are not whole numbers since 900 is not divisible by seven.

Formula for calculating the area of a heptagon

The area of a regular heptagon is determined only by the length of its sides. In a standard formula, one multiplies the square of the side length by a constant that is calculated from the cotangent of one-seventh of an angle corresponding to half a circle.

Here A is the area and s is the length of a side. Since the cotangent of an angle of about 25.71 degrees is approximately 2.0765, the entire constant by which the square of the side length is multiplied is roughly 3.634. This means that the area of a regular heptagon is approximately equal to the product of the side length squared and further multiplied by 3.634.

If the radius of the inscribed circle or the radius of the circumscribed circle is known instead of the side length, then the area can be calculated in the same way using r. The former is the distance from the center to the midpoint of a side, while the latter is the distance from the center to one of the vertices.

Instructions for using this calculator:

Enter any measurement such as side length, perimeter, inradius, circumradius or area into the appropriate input field and select a unit like centimeter or inch. Once you have entered one value, the calculator will automatically fill out all other measurements and display the area as the most important result. It is not necessary to specify which type of measurement you have entered.

If you want to repeat the calculations with other values, first delete the fields that contain the previous values and then enter new values. By opening the comparison option you can see the circle around the heptagon and the circle inside of the heptagon. When you open the graph you can observe how the area changes as the side length becomes longer.

List of all measurements for the heptagon

Each measurement of a regular heptagon is connected to the other measurements by the length of its sides. The following table shows the relationships used by the calculator and explains why only one value needs to be entered in order to calculate the remaining values.

These constants do not contain simple integers or the golden ratio as is the case with pentagons and decagons. The numerical values of the heptagon remain stubbornly irrational, which ties in to a known fact that a regular heptagon cannot be constructed using only compasses and straightedges.

A complete example

Consider a regular heptagon with side length of 10 centimeters.

We square the side length to get 100, and multiply that by the constant for heptagons.

So the area of a heptagon with side length of 10 centimeters is approximately 363.39 square centimeters. The perimeter of that same heptagon would be 70 centimeters, and its inradius would be about 10.38 centimeters while its circumradius would be around 11.52 centimeters. You can find all these results directly with this calculator by simply entering the value 10 into the side length field.

Examples of area for common side lengths:

The following table shows the areas for some common side lengths, with results rounded to two decimal places. Note that the area increases rapidly as the length of a side is squared.

Calculation method via the radius of the inscribed circle: division into seven triangles

There is another way to understand the area that applies to all regular polygons. Drawing a line from the center to each vertex completely divides the heptagon into seven identical triangles. Since the base of each triangle is equal to the side length and the height is equal to the inradius, the area is one-half the perimeter times the inradius.

This explains why the radius of the inscribed circle is so useful. If you can measure the distance from the center to the midpoint of a side, then you don't have to square the length of the sides in order to calculate the area. For this reason, this calculator accepts the radius of the inscribed circle as an input value.

A heptagon and two circles

Any regular heptagon will fit snugly between two circles. The larger circle passes through the seven vertices, and its radius is equal to that of the circumcircle. The smaller circle touches the midpoints of each side, and its radius is equal to that of the incircle.

As the shape of the figure is constant, its area will always be approximately 87.1 percent of the area of the larger circle and about 7.3 percent greater than the area of the smaller circle. These proportions do not change even if size varies. It is natural that there is slightly more free space inside the circumcircle of a heptagon than in the case of a decagon, as the lower number of sides means that the shape deviates slightly from being a circle. If you open the comparison options above, you can find out the areas of the two circles corresponding to your heptagon.

Diagonals of a heptagon

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

There are only two different lengths for these diagonals; the shorter ones cross one vertex while the longer ones cross two vertices. Because the seven 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 points as shown by the calculator.

In a heptagon there are two different lengths and between these there is another interesting relationship. If you take the side length, short diagonal and long diagonal then their reciprocals add up to exactly one.

This simple identity is characteristic of the regular heptagon and appears repeatedly in its geometry.

Why the heptagon is special:

The regular heptagon is the smallest regular polygon that cannot be constructed with a compass and straightedge without markings. Triangles, squares, pentagons, and hexagons can all be constructed in this way, as can decagons but not heptagons. Mathematicians have proven this using the algebra of constructible numbers, which also explains why the angles and ratios of the heptagon are often irrational.

In practice a regular heptagon is either constructed using a protractor by using the 51.43-degree central angle seven times in succession, or else it uses a very accurate approximation construction. This calculator completely bypasses this problem and calculates based on exact trigonometric functions.

Where does the heptagon come in?

Seven-sided patterns are more common than you might think. Both the British fifty pence and twenty pence coins are seven-sided, with a slightly curved edge. These symmetrical shapes roll smoothly in vending machines, while at the same time providing a shape that people can easily distinguish by touch. Other countries have also minted coins with seven sides for similar reasons.

Architects use seven-sided floor plans for towers and pavilions while designers incorporate heptagonal contours into logos and signs. The shape is also found in puzzles and occasionally the battlements of fortresses. In teaching, the regular heptagon is often used as an example precisely because its values are not "nice".

Tips for accurate results:

Before using the length of a side to find area, make sure that your heptagon is actually regular, or has equal sides and angles. The formula requires that all seven sides and angles be the same for it to work. Be consistent with units. If you use centimeters to measure the length of a side, expect your answer in square centimeters.

Remember that small errors in the length of a side will lead to larger errors in the calculated area since the area is proportional to the square of the length of a side. If possible also measure and compare the radius of the inscribed or circumscribed circle and enter these values as well to make sure that the result is correct for the same heptagon.