Decagon

What Is a Decagon?

A decagon is a polygon with exactly 10 sides, 10 vertices (corners), and 10 interior angles. The name originates from the Greek words "deka," meaning ten, and "gonia," meaning angle. Like other polygons, decagons can be classified in several ways, giving us a more detailed picture of their characteristics.

We can categorize decagons into two main types:

• Regular Decagon: This is a decagon where all 10 sides are equal in length, and all 10 interior angles are equal in measure. When people simply say "decagon" in a geometry context, they are often referring to a regular decagon because of its perfect symmetry.
• Irregular Decagon: This is a decagon where the sides have different lengths, and the interior angles have different measures. They can come in countless shapes and sizes.

Furthermore, decagons can also be described as convex or concave:

• Convex Decagon: All interior angles are less than ${180}^{\circ }$. If you draw a line segment connecting any two vertices, the entire segment will lie inside the decagon. Regular decagons are always convex.
• Concave Decagon: At least one interior angle is greater than ${180}^{\circ }$ (a reflex angle). This results in a shape that appears to have a "dent" or be caved in.

In this lesson, we will primarily focus on the properties and calculations related to regular convex decagons, as they are the most commonly studied type.

What Are the Key Properties of a Regular Decagon?

A regular decagon is a highly symmetrical and predictable shape. Its properties are consistent, which makes it easy to perform calculations. Here are the most important characteristics:

1. Equal Sides: All 10 sides have the same length, which we often denote with the variable $s$.
2. Equal Angles: All 10 interior angles are congruent, each measuring ${144}^{\circ }$. Similarly, all 10 exterior angles are congruent, each measuring ${36}^{\circ }$.
3. Convexity: As mentioned, all regular decagons are convex. None of their vertices point inward.
4. Diagonals: A regular decagon has 35 diagonals of four different lengths.
5. Symmetry: It possesses a high degree of symmetry. It has 10 lines of reflectional symmetry and rotational symmetry of order 10 (meaning it looks the same after being rotated by multiples of ${36}^{\circ }$ around its center).
6. Inscribable and Circumscribable: A regular decagon can be perfectly inscribed in a circle (a circumcircle) that touches all its vertices, and a circle (an incircle) can be perfectly inscribed within it, touching the midpoint of each side. The center of these two circles is the same.

These properties form the foundation for deriving the formulas used to calculate a decagon's angles, area, and other attributes.

How Do You Calculate the Angles of a Decagon?

Understanding the angles of a polygon is a fundamental skill in geometry. For any decagon, regular or irregular, the sum of its interior angles is always the same. We can find this using a general formula for any polygon with $n$ sides.

Sum of the Interior Angles

The formula to find the sum of the interior angles of any convex polygon is:

Here, $S$ is the sum and $n$ is the number of sides. For a decagon, $n=10$. Let's plug this into the formula:

$$S=(10-2)\times {180}^{\circ }$$
$$S=8\times {180}^{\circ }$$
$$S={1440}^{\circ }$$

So, the sum of all interior angles in any convex decagon is ${1440}^{\circ }$.

Measure of a Single Interior Angle (Regular Decagon)

For a regular decagon, all 10 interior angles are equal. To find the measure of a single angle, we simply divide the total sum by the number of angles (which is 10):

Each interior angle in a regular decagon is exactly ${144}^{\circ }$.

Exterior Angles

The sum of the exterior angles of any convex polygon is always ${360}^{\circ }$. For a regular decagon, since all exterior angles are equal, we can find the measure of one by dividing the total by 10:

$$\text{Exterior Angle}=\frac{{360}^{\circ }}{n}=\frac{{360}^{\circ }}{10}={36}^{\circ }$$

Notice that an interior angle and its corresponding exterior angle are supplementary, meaning they add up to ${180}^{\circ }$. We can check our work: ${144}^{\circ }+{36}^{\circ }={180}^{\circ }$.

How Do You Find the Area of a Regular Decagon?

Calculating the area of a regular decagon can seem intimidating, but it's straightforward once you have the right formula. There are a few different formulas depending on what information you are given: the side length ($s$), the apothem ($a$), or the circumradius ($R$).

The apothem is the distance from the center of the polygon to the midpoint of a side, forming a right angle with that side. The circumradius is the distance from the center to any vertex.

Area Using Side Length (s)

This is the most common formula when you only know the length of one side. It is derived using trigonometry by dividing the decagon into 10 congruent isosceles triangles.

While it looks complex, you can approximate the radical part: $\sqrt{5+2\sqrt{5}}\approx 3.07768$. This simplifies the formula to:

$$\text{Area}\approx \frac{5}{2}{s}^2(3.07768)\approx 7.6942s^2$$

Area Using Apothem (a) and Perimeter (P)

A more general formula for any regular polygon is:

Since the perimeter $P$ of a regular decagon is $10s$, the formula is $\text{Area}=\frac{1}{2}\times (10s)\times a=5sa$.

How Many Diagonals Does a Decagon Have?

A diagonal is a line segment that connects two non-adjacent vertices of a polygon. Counting them one by one would be tedious and prone to error. Fortunately, there is a simple formula to calculate the number of diagonals for any polygon with $n$ sides.

Let's break down why this formula works:

• From each vertex (there are $n$ of them), you can draw a diagonal to every other vertex except for itself and its two immediate neighbors (the vertices it shares a side with). That's why we use $(n-3)$.
• If we multiply $n\times (n-3)$, we are counting each diagonal twice (once from each endpoint). For example, the diagonal from vertex A to vertex D is the same as the diagonal from D to A.
• To correct for this double-counting, we divide the result by 2.

For a decagon, we have $n=10$. Let's apply the formula:

$$D=\frac{10(10-3)}{2}$$
$$D=\frac{10(7)}{2}$$
$$D=\frac{70}{2}$$
$$D=35$$

A decagon has exactly 35 diagonals.

What Are Common Mistakes When Working with Decagons?

When learning about a new shape, it's easy to make small mistakes. Being aware of these common pitfalls can help you avoid them.

• Confusing Polygon Names: Students often mix up decagons (10 sides) with octagons (8 sides) or dodecagons (12 sides). It's helpful to remember the Greek prefixes: octa (8), deca (10), dodeca (12).
• Applying Regular Formulas to Irregular Shapes: The formulas for a single interior angle (${144}^{\circ }$) and the area based on side length only work for regular decagons. The sum of interior angles (${1440}^{\circ }$) and the number of diagonals (35) apply to any convex decagon.
• Errors in the Diagonals Formula: A common mistake is forgetting to divide by 2 in the formula $\frac{n(n-3)}{2}$. This leads to an answer that is exactly double the correct number.
• Mixing up Apothem and Radius: When using area formulas, it's crucial to distinguish between the apothem (center to midpoint of a side) and the circumradius (center to a vertex). They are not the same length.
• Calculation Errors: The area formula $\frac{5}{2}{s}^2\sqrt{5+2\sqrt{5}}$ is complex. Be careful when entering it into a calculator, paying close attention to parentheses and the order of operations, especially with the nested square root.

Decagon Formulas: A Quick Summary

Here is a quick reference table summarizing the key formulas related to decagons. This can be a helpful study guide.