Pentagon

What Is a Pentagon?

A pentagon is a two-dimensional polygon that has exactly five straight sides and five vertices (corners). The name itself gives us a clue: it comes from the Greek words pente, meaning "five," and gonia, meaning "angle." So, a pentagon is literally a "five-angle" figure. To qualify as a pentagon, a shape must be a closed figure with sides made of straight lines. A shape with any curved sides or any openings is not a polygon, and therefore cannot be a pentagon.

Every simple pentagon (one that doesn't intersect itself) has several key features:

• Five Sides: These are the straight line segments that form the boundary of the pentagon.
• Five Vertices: A vertex is a point where two sides meet. A pentagon has five of these corners.
• Five Interior Angles: These are the angles inside the pentagon at each vertex. The sum of these angles is always the same for any simple pentagon, a property we will explore later.

Pentagons are all around us. The most famous example is the Pentagon building in Arlington, Virginia, which houses the headquarters of the U.S. Department of Defense. You can also find them in the pattern on a soccer ball (along with hexagons) and in the shape of home plate in baseball.

What Are the Different Types of Pentagons?

Not all pentagons look the same. We can classify them into different categories based on their sides, angles, and overall shape. The main distinctions are between regular and irregular, and convex and concave.

Regular vs. Irregular Pentagons

• A regular pentagon is the most symmetrical type. It is equilateral (all five sides have the same length) and equiangular (all five interior angles are equal). Because of this uniformity, it's the classic, star-like shape many people picture first.
• An irregular pentagon is any pentagon that is not regular. This means that its sides can have different lengths, and its angles can have different measures. The vast majority of pentagons you can draw are irregular.

Convex vs. Concave Pentagons

• A convex pentagon is a pentagon where all interior angles are less than ${180}^{\circ }$. If you were to draw a line connecting any two vertices, the line would stay entirely inside the pentagon. It has no "dents" or parts that cave inward. All regular pentagons are convex.
• A concave pentagon has at least one interior angle that is greater than ${180}^{\circ }$. This large angle creates a "dent" in the shape, making it look like it has caved in on itself. If you connect two vertices in a concave pentagon, the line segment might go outside the shape.

Here is a table summarizing these types:

How Do You Calculate the Angles of a Pentagon?

One of the most important properties of any polygon is the sum of its interior angles. There's a reliable formula for this that works for any polygon, including pentagons.

Sum of Interior Angles

The formula to find the sum of the interior angles of a polygon with $n$ sides is:

For a pentagon, we know that $n=5$. Let's plug this into the formula:

This is a crucial fact: the sum of the interior angles in any simple pentagon is always ${540}^{\circ }$. This is true whether the pentagon is regular, irregular, convex, or concave.

Angles in a Regular Pentagon

In a regular pentagon, all five interior angles are equal. To find the measure of a single angle, we simply take the total sum (${540}^{\circ }$) and divide it by the number of angles ($5$):

So, every interior angle in a regular pentagon measures exactly ${108}^{\circ }$.

Exterior Angles

An exterior angle is the angle formed by one side of a polygon and the extension of an adjacent side. For any convex polygon, the sum of the exterior angles is always ${360}^{\circ }$. For a regular pentagon, we can find the measure of one exterior angle by dividing the total by $5$:

Notice that an interior angle and its corresponding exterior angle are supplementary, meaning they add up to ${180}^{\circ }$. For a regular pentagon, this holds true: ${108}^{\circ }+{72}^{\circ }={180}^{\circ }$.

Calculating the Perimeter of a Pentagon

The perimeter of any polygon is the total distance around its exterior. To find the perimeter, you simply add up the lengths of all its sides. This concept is the same for all polygons, from triangles to pentagons and beyond.

Perimeter of an Irregular Pentagon

For an irregular pentagon, the sides can have different lengths. Let's label the lengths of the five sides as $s_1,s_2,s_3,s_4,$ and $s_5$. The formula for the perimeter $P$ is:

You just need to measure each side and find their sum.

Perimeter of a Regular Pentagon

For a regular pentagon, the calculation is much simpler. By definition, all five sides are equal in length. If we let $s$ represent the length of one side, then the perimeter is just $5$ times $s$:

How Do You Find the Area of a Regular Pentagon?

Calculating the area of an irregular pentagon is complex and often involves dividing it into triangles and finding the area of each. However, for a regular pentagon, there is a straightforward formula. This formula requires a special measurement called the apothem.

What is an Apothem?

The apothem (usually denoted by $a$) of a regular polygon is the distance from the center of the polygon to the midpoint of any side. It is a line segment that is perpendicular to the side. You can visualize a regular pentagon as being made up of five identical isosceles triangles, with their bases forming the sides of the pentagon. The apothem is the height of each of these triangles.

The Area Formula

The area of any regular polygon can be found using the perimeter and the apothem. The general formula is:

Since the perimeter $P$ of a regular pentagon is $5s$, we can substitute that into the formula:

Where:

• $A$ is the area.
• $s$ is the length of a side.
• $a$ is the length of the apothem.

In higher-level math, you can calculate the apothem from the side length using trigonometry, but in 7th and 8th grade, the apothem is usually given in the problem.

Properties of Diagonals in a Pentagon

A diagonal is a line segment that connects two non-adjacent vertices of a polygon. Diagonals can reveal interesting properties about a shape's internal structure.

How Many Diagonals Does a Pentagon Have?

We can count them by drawing them, but there is also a formula to calculate the number of diagonals $D$ for any polygon with $n$ sides:

For a pentagon, $n=5$. Let's apply the formula:

A pentagon has exactly 5 diagonals. If you draw all 5 diagonals inside a regular pentagon, they form a five-pointed star, with a smaller pentagon in the center. This shape is known as a pentagram.

The Golden Ratio

One of the most fascinating aspects of a regular pentagon is its relationship with the golden ratio, an irrational number approximately equal to $1.618$ and often represented by the Greek letter phi ($\varphi$). In a regular pentagon, the ratio of the length of a diagonal to the length of a side is equal to the golden ratio.

This remarkable property has made the pentagon and pentagram significant symbols in art, architecture, and mysticism for centuries.

Common Mistakes to Avoid

When working with pentagons, students sometimes make predictable errors. Being aware of these can help you avoid them.

1. Assuming All Pentagons are Regular: A common mistake is to assume that any five-sided figure has angles of ${108}^{\circ }$. This is only true for regular pentagons. For irregular pentagons, you must use the fact that the angles sum to ${540}^{\circ }$ and solve for what's missing.
2. Confusing Perimeter and Area: Remember that perimeter is the distance around a shape (measured in linear units like cm, m, ft) while area is the space inside the shape (measured in square units like cm${}^2$, m${}^2$, ft${}^2$). Don't mix up their formulas or units.
3. Incorrectly Calculating the Sum of Angles: Students sometimes forget the $(n-2)$ part of the angle sum formula and just multiply $n\times {180}^{\circ }$. Always subtract $2$ from the number of sides before multiplying by ${180}^{\circ }$.
4. Using the Area Formula for Irregular Pentagons: The formula $A=\frac{5}{2}sa$ works only for regular pentagons because it relies on the apothem and equal side lengths. It cannot be applied to irregular shapes.
5. Mixing Up Apothem and Radius: The apothem goes from the center to the midpoint of a side. The radius of a regular polygon goes from the center to a vertex. They have different lengths and are used in different calculations.

Quick Summary and Key Formulas

Here is a quick reference guide to the most important concepts and formulas related to pentagons.

Key Properties:

• A pentagon is a polygon with $5$ sides, $5$ vertices, and $5$ angles.
• Regular Pentagon: All sides are equal, and all interior angles are equal (${108}^{\circ }$).
• Irregular Pentagon: Sides and/or angles are not all equal.
• Convex Pentagon: All interior angles are less than ${180}^{\circ }$.
• Concave Pentagon: At least one interior angle is greater than ${180}^{\circ }$.
• A pentagon has $5$ diagonals.

Key Formulas:

• Sum of Interior Angles: $S=(n-2)\times {180}^{\circ }=(5-2)\times {180}^{\circ }={540}^{\circ }$
• Each Interior Angle (Regular): $\frac{{540}^{\circ }}{5}={108}^{\circ }$
• Sum of Exterior Angles: ${360}^{\circ }$
• Each Exterior Angle (Regular): $\frac{{360}^{\circ }}{5}={72}^{\circ }$
• Perimeter (Irregular): $P=s_1+s_2+s_3+s_4+s_5$
• Perimeter (Regular): $P=5s$
• Area (Regular): $A=\frac{5}{2}sa$ (where $a$ is the apothem)
• Number of Diagonals: $D=\frac{n(n-3)}{2}=\frac{5(5-3)}{2}=5$