**What is a heptagon?**

He **heptagon** is a polygon with seven sides and seven interior angles. As a geometric word, the word heptagon originates from the Greek *hepta*which means seven, and *gonos*, which translates as angle. It is, then, a polygon with seven angles.

A polygon is a flat geometric figure that is formed by joining and closing three or more segments, also called *sides*. The points in common that the sides have are called *vertices*.

The region between adjacent sides, on the inside of the figure, is the *internal angle*whose vertex is also one of the vertices of the heptagon.

If all sides and interior angles have the same measure, it is a *regular heptagon*otherwise it is a *irregular heptagon*. Irregular heptagons take a variety of shapes.

A heptagon can also be *convex* either *concave*according to the measure of its internal angles. If the internal angles are less than 180°, the heptagon is convex, but if one or more of these angles is greater than 180°, then it is concave.

A heptagon whose sides all have the same length is called *equilateral*. This can be concave or convex, regular or irregular.

**regular and irregular heptagon**

The regular heptagon is one that has its seven sides and its seven internal angles of equal measure, the opposite of an irregular heptagon, which has at least one side of a different measure, or a different internal angle.

**the regular heptagon**

**Symmetry**

A regular heptagon is a highly symmetric figure. Line segments can be drawn connecting a vertex to the midpoint of the opposite side, all of which intersect at the center of the heptagon. These are the seven axes of symmetry of the figure.

The segment that joins a vertex with the center of the heptagon is called *circumradius*since it corresponds to the radius of the single circle that passes through each and every one of the vertices, as shown in the figure.

**angles**

The following angles stand out in the heptagon:

**internal angle ****ϕ**, whose vertex coincides with one of the vertices of the heptagon, the sides of the angle being two of the adjacent sides of the heptagon. For a regular heptagon, the measure of each of the seven interior angles is approximately 128.57°.

**external angle**, the one formed between one of the sides and the extension of one of the adjacent sides, the common vertex between these two sides being the vertex of the angle. Likewise, seven external angles are formed and their value is calculated by subtracting 180° from the internal angle, which for the regular heptagon results in 51.43°.

**central angle ****θ**, has its vertex in the center of the regular heptagon, and its sides are the radii of the heptagon, that is, the segments that connect the center with each vertex, as shown in the following figure. It is calculated by dividing 360° by 7, which gives an approximate result of 51.43°.

The sum of the internal angle and the central angle is equal to 180°, that is:

ϕ + θ = π

**Area**

For the regular heptagon there are formulas, while for the irregular heptagon you have to resort to other methods, such as dividing it into other simpler polygons, such as triangles.

**Area of the regular heptagon**

**1. If the perimeter P and the apothem ap are known:**

Let A be the area of the heptagon. There is a formula to calculate the area, valid for any regular polygon:

**2. If the side L and the apothem are known ****app:**

Since the perimeter is the sum of the sides, and the side measures L in the regular heptagon, we get:

P = 7⋅L

Substituting in the previous formula:

**3. If side L is known **

The following is an approximate formula, valid when the measure of the side L of the heptagon is known:

A = 3.634∙L2

**Area of the irregular heptagon**

The area of the irregular heptagon can be calculated by *triangulation*, which consists of dividing the heptagon into five triangles (see the following figure). The area of each one is calculated and the results are added, thus obtaining the complete area of the heptagon.

The other method is called *Gaussian determinant*, and it is required to place the heptagon in a rectangular coordinate system, in order to know the coordinates of each vertex. The area is calculated by a formula involving the values of these coordinates.

**diagonals**

The *diagonals *are segments that connect a vertex with another that is not consecutive (if the segment connects two consecutive vertices it is the *side*). A heptagon has 14 diagonals in all.

The number of diagonals D of any polygon with n sides is calculated using the formula:

D = n · (n – 3) / 2

Substituting n = 7, it remains:

D = 7 · (7 – 3) / 2 = 7 · (4 / 2) = 14

**sum of interior angles**

For any heptagon, regardless of whether it is regular or not, the sum of its internal angles is equal to 900° or 5π radians.

This property is very easy to demonstrate, for this the heptagon is divided into individual triangles that do not overlap, drawing rectilinear segments that join the vertices, without intersecting each other.

Five triangles are obtained and in each one, the sum of their internal angles is 180°, which multiplied by 5 equals 900°:

5 x 180° = 900°

**formulas**

**Perimeter**

For a regular heptagon with side L, the perimeter P is calculated as follows:

P = 7⋅L

If the perimeter is irregular, add the lengths of each of the seven sides.

**internal angle measure**

In a regular heptagon, the internal angle θ measures:

θ = [180 (n-2)]/n

Where n = 7.

**Apothem**

Let L be the side of the regular heptagon. The apothem is the segment that goes from the center of the heptagon, perpendicularly to the opposite side.

Let ap be the length of the apothem. Knowing the radius of the circumscribed circle, which is denoted as rc and the side L of the heptagon, we have:

Knowing the internal angle θ, the above is equivalent to:

**Area**

If it is a regular heptagon with side L, the area A is given by:

A = 3.634∙L2

When the heptagon is irregular, the rectangular coordinates of each vertex are needed, given by (xn , yn), where n = 1, 2, 3… 7.

Then the following formula is applied to find the area A:

**diagonals**

The number D of diagonals is given by:

D = n · (n – 3) / 2

Where n = 7 for the heptagon.

**How to make a heptagon**

The following animation shows how to roughly draw a regular heptagon, using a ruler and compass.

**References**

Alexander, D. 2013. Geometry. 5th. Edition. Cengage Learning.

Lemonis, M. Regular heptagon calculator. Retrieved from: calcresource.com.

Math Open Reference. Area of a polygon. Retrieved from: mathopenref.com.

Universe Formulas. Heptagon. Recovered from: universoformulas.com.

Wikipedia. Heptagon. Recovered from: es.wikipedia.com.