Heptagon

What Exactly Is a Heptagon?

A heptagon is a two-dimensional polygon with seven straight sides and seven angles. The name originates from the Greek words "hepta," meaning seven, and "gonia," meaning angle. Every heptagon also has seven vertices, which are the corner points where the sides meet. Like other polygons, heptagons can be categorized based on the properties of their sides and angles.

We can classify heptagons into two main types:

• Regular Heptagon: A regular heptagon is one where all seven sides are equal in length (equilateral) and all seven interior angles are equal in measure (equiangular). When people talk about a "heptagon" without any other description, they are usually referring to a regular heptagon.
• Irregular Heptagon: An irregular heptagon is a seven-sided polygon where the sides are not all the same length, and the angles are not all the same measure. They can come in countless different shapes.

Furthermore, heptagons can be either convex or concave:

• Convex Heptagon: This is the most common type. All its interior angles are less than ${180}^{\circ }$, and all vertices point outwards. If you draw a line between any two vertices, the line will stay entirely inside the shape. All regular heptagons are convex.
• Concave Heptagon: A concave heptagon has at least one interior angle that is greater than ${180}^{\circ }$. This creates a "dent" or an indentation in the shape, where at least one vertex points inward.

What Are the Key Properties of Heptagons?

All heptagons, whether regular or irregular, share a few fundamental characteristics simply because they are seven-sided polygons. However, the regular heptagon has a much longer and more specific list of properties due to its symmetry.

Properties of All Convex Heptagons

• They have exactly $7$ sides, $7$ vertices, and $7$ interior angles.
• The sum of the interior angles is always ${900}^{\circ }$.
• The sum of the exterior angles is always ${360}^{\circ }$.
• The shape can be divided into $5$ triangles by drawing diagonals from a single vertex.
• It has $14$ diagonals in total.

Specific Properties of a Regular Heptagon

A regular heptagon is highly symmetrical, which gives it several additional, predictable properties:

• Congruent Sides: All $7$ sides are equal in length.
• Congruent Angles: All $7$ interior angles are equal, each measuring approximately ${128.57}^{\circ }$.
• Equal Exterior Angles: All $7$ exterior angles are equal, each measuring approximately ${51.43}^{\circ }$.
• Symmetry: It has $7$ lines of reflectional symmetry. Each line passes through a vertex and the midpoint of the opposite side.
• Rotational Symmetry: It has rotational symmetry of order $7$. This means you can rotate it by increments of $\frac{{360}^{\circ }}{7}$ around its center, and it will look identical at each step.
• Cyclic: A regular heptagon is a cyclic polygon, meaning all its vertices lie on a single circle, called the circumcircle.

How Do You Calculate the Angles of a Heptagon?

Calculating the angles of a heptagon relies on a general formula that applies to all convex polygons. The key is knowing the number of sides, which for a heptagon is always $n=7$.

Sum of the Interior Angles

To find the sum of all interior angles in any convex polygon, you can use the following formula:

Here, $n$ is the number of sides. For a heptagon, we substitute $n=7$:

This is a crucial property: the interior angles of any simple, convex heptagon will always add up to ${900}^{\circ }$, regardless of whether it's regular or irregular.

Measure of a Single Interior Angle (Regular Heptagon)

For a regular heptagon, all seven interior angles are equal. To find the measure of just one of these angles, we take the total sum (${900}^{\circ }$) and divide it by the number of angles ($7$).

Applying this to our heptagon:

Each interior angle in a regular heptagon is approximately $128.57$ degrees.

Exterior Angles

An exterior angle is formed by extending one side of the polygon and measuring the angle between that extension and the adjacent side. For any convex polygon, the sum of all exterior angles is always ${360}^{\circ }$.

In a regular heptagon, since all exterior angles are equal, we can find the measure of one by dividing ${360}^{\circ }$ by $7$:

Notice that an interior angle and its corresponding exterior angle form a straight line, so they add up to ${180}^{\circ }$. We can check our work: ${128.57}^{\circ }+{51.43}^{\circ }={180}^{\circ }$.

How Many Diagonals Does a Heptagon Have?

A diagonal is a line segment that connects two non-adjacent vertices of a polygon. You could try to draw all the diagonals in a heptagon and count them, but this can get messy and it's easy to miss one or count one twice. A much more reliable method is to use the general formula for finding the number of diagonals in a polygon with $n$ sides.

Let's break down why this formula works:

• From any single vertex, you can draw a diagonal to every other vertex except for itself and its two immediate neighbors (the adjacent vertices). So, from one vertex, you can draw $(n-3)$ diagonals.
• Since there are $n$ vertices, you might think the total is $n(n-3)$.
• However, this method counts each diagonal twice (once from each endpoint, e.g., from vertex A to vertex C and again from C to A). Therefore, we must divide the result by $2$.

Now, let's apply this formula to the heptagon, where $n=7$:

A heptagon has exactly $14$ diagonals. This is true for both regular and irregular convex heptagons.

How to Find the Perimeter and Area of a Regular Heptagon?

Calculating the perimeter of a heptagon is straightforward. Finding its area, however, requires a bit more geometry. The formulas below apply specifically to regular heptagons.

Perimeter of a Regular Heptagon

The perimeter is the total distance around the outside of the polygon. Since a regular heptagon has $7$ sides of equal length, you can find its perimeter by multiplying the length of one side ($s$) by $7$.

For an irregular heptagon, you would simply add the lengths of all seven different sides.

Area of a Regular Heptagon

There are two common methods to find the area of a regular heptagon.

Method 1: Using the Apothem

The apothem ($a$) is a line segment from the center of a regular polygon to the midpoint of a side. It is also the height of the isosceles triangle formed by connecting two adjacent vertices to the center. A regular heptagon can be divided into $7$ such identical triangles.

The area of one of these triangles is $\frac{1}{2}\times \text{base}\times \text{height}$, which is $\frac{1}{2}\times s\times a$. Since there are $7$ triangles, the total area is:

We can also write this using the perimeter ($P=7s$):

Method 2: Using Only the Side Length

If you don't know the apothem, you can still find the area using a more complex formula derived from trigonometry. While the full derivation is advanced, the resulting formula is very useful.

This looks intimidating, but the $\cot (\frac{{180}^{\circ }}{7})$ part is just a constant value. Its approximate value is $2.0765$. This gives us a much simpler, practical formula for calculations:

Can You Draw a Perfect Heptagon?

In the world of classical geometry, "to construct" a shape means to draw it perfectly using only two tools: an unmarked straightedge (a ruler without markings) and a compass. You can construct shapes like equilateral triangles, squares, pentagons, and hexagons this way. But what about a regular heptagon?

The surprising answer is no. It is mathematically impossible to construct a perfect regular heptagon using only a compass and straightedge. This was part of a larger discovery by the famous mathematician Carl Friedrich Gauss in the late 18th century. He determined exactly which regular polygons are constructible and which are not. The heptagon, with its $7$ sides, falls into the "impossible" category.

This doesn't mean we can't draw one! In practice, artists, engineers, and designers use various methods to create highly accurate approximations of a regular heptagon. These methods might involve using a protractor to measure the $\approx {128.57}^{\circ }$ angles or using special geometric techniques that get incredibly close to the true shape. So while a perfect, classical construction is impossible, creating a visually perfect heptagon for all practical purposes is very achievable.

Common Mistakes to Avoid with Heptagons

When working with heptagons, students sometimes fall into common traps. Being aware of these can help you avoid losing points on homework and tests.

• Mixing up $n$: The most frequent error is using the wrong value for $n$. Always remember that for a heptagon, $n=7$. It's easy to accidentally use $n=6$ (for a hexagon) or $n=8$ (for an octagon) in formulas if you're in a hurry.
• Confusing Angle Sum with Single Angle: Don't mix up the sum of the interior angles (${900}^{\circ }$) with the measure of a single interior angle in a regular heptagon ($\approx {128.57}^{\circ }$). The first applies to all heptagons, the second only to regular ones.
• Applying Regular Formulas to Irregular Shapes: The formulas for a single interior angle, perimeter ($P=7s$), and area ($A\approx 3.634s^2$) only work for regular heptagons. You cannot use them if a heptagon has sides or angles of different measures.
• Diagonal Formula Errors: When calculating diagonals with $\frac{n(n-3)}{2}$, a common mistake is forgetting to divide by $2$. Remember, you must divide by two to correct for double-counting each diagonal.
• Area Calculation Mix-ups: When using the apothem formula $A=\frac{1}{2}Pa$, ensure you are using the apothem (center to midpoint of a side) and not the radius (center to a vertex). They are different lengths.

Heptagon Formulas: A Quick Reference Guide

Here is a summary of the most important properties and formulas for a heptagon, especially a regular heptagon, in one place.