Area of a Pentagon: Formulas, Calculations, and Examples

A pentagon is a five-sided polygon, and calculating its area depends on whether it is a regular pentagon (all sides and angles are equal) or an irregular pentagon (sides and angles are not necessarily equal). The calculation methods for the area of a pentagon vary depending on the type, but each approach is manageable with basic geometry and algebra. This article explains different methods to calculate the area of a pentagon, offering detailed examples for both regular and irregular pentagons.

1. Basic Understanding of Pentagon Geometry

A pentagon is any five-sided polygon, meaning it has five straight sides and five interior angles. Pentagons are commonly classified into two types:

Regular Pentagon: All five sides and interior angles are equal.
Irregular Pentagon: Sides and angles may vary, with no uniform structure.

In both cases, the sum of the interior angles of a pentagon is always:

$(5 - 2) \times 180^\circ = 540^\circ$

This is true for any pentagon, regardless of whether it’s regular or irregular. However, the formula for calculating the area differs depending on the type of pentagon.

2. Area of a Regular Pentagon

The simplest way to calculate the area of a pentagon is for a regular pentagon. In this case, all five sides are equal in length, and each interior angle measures $108^\circ$ . The formula to find the area of a regular pentagon with side length $s$ is derived using basic trigonometric principles or by dividing the pentagon into five identical triangles.

Formula for Area of a Regular Pentagon

The area ( $A$ ) of a regular pentagon with side length $s$ is given by:

$A = \frac{1}{4} \sqrt{5 (5 + 2 \sqrt{5})} \cdot s^2$

This formula allows you to calculate the area directly from the side length without needing any additional measurements.

Alternatively, another method involves calculating the apothem (the line segment from the center of the pentagon perpendicular to one of its sides). This approach breaks down the pentagon into five isosceles triangles.

Formula Using Apothem

If you know the apothem $a$ (distance from the center to the midpoint of a side) and the side length $s$ , you can use the following formula:

$A = \frac{5 \times s \times a}{2}$

This formula is useful in cases where the apothem is given or easier to measure than the side length.

Example 1: Calculating the Area of a Regular Pentagon Using Side Length

Suppose you have a regular pentagon with a side length of $s = 6$ units. Using the area formula for a regular pentagon:

$A = \frac{1}{4} \sqrt{5 (5 + 2 \sqrt{5})} \cdot s^2$

Substitute $s = 6$ :

$A = \frac{1}{4} \sqrt{5 (5 + 2 \sqrt{5})} \cdot 6^2$

$A = \frac{1}{4} \cdot \sqrt{5 (5 + 2 \sqrt{5})} \cdot 36$

Calculating further, you would find:

$A \approx 61.94 \text{ square units}$

This calculation provides the area of the pentagon.

Example 2: Calculating the Area Using the Apothem

Suppose we have a regular pentagon with side length $s = 10$ units and an apothem $a = 6.88$ units.

Using the apothem-based formula:

$A = \frac{5 \times s \times a}{2}$

$A = \frac{5 \times 10 \times 6.88}{2}$

$A = \frac{5 \times 68.8}{2}$

$A = \frac{344}{2}$

$A = 172 \text{ square units}$

This method is straightforward when you know both the side length and the apothem of the pentagon.

3. Area of an Irregular Pentagon

For an irregular pentagon, the calculation process is slightly more complex because the sides and angles vary. The most effective approach is to divide the pentagon into triangles, calculate the area of each triangle separately, and then sum these areas.

Steps to Calculate the Area of an Irregular Pentagon

1. Divide the Pentagon into Triangles: Draw diagonals within the pentagon from one vertex to non-adjacent vertices to create non-overlapping triangles.
2. Calculate the Area of Each Triangle: Use the triangle area formula for each triangle. If you know the base and height of a triangle, use:

$\text{Area of Triangle} = \frac{1}{2} \times \text{base} \times \text{height}$

If you know the coordinates of the triangle’s vertices, you can use the shoelace formula to find the area.
3. Sum the Areas of All Triangles: Add up the areas of the triangles to get the total area of the pentagon.

Example: Calculating the Area of an Irregular Pentagon

Suppose we have an irregular pentagon with vertices at the following coordinates:

$A (0, 0)$
$B (4, 0)$
$C (5, 3)$
$D (2, 5)$
$E (-1, 3)$

1. Divide the Pentagon into Triangles: Draw diagonals to form triangles. For this example, we’ll create three triangles: $\triangle ABC$ , $\triangle ACD$ , and $\triangle ADE$ .

2. Calculate Each Triangle’s Area Using the Shoelace Formula:

The shoelace formula for a triangle with vertices $(x_1, y_1)$ , $(x_2, y_2)$ , and $(x_3, y_3)$ is:

$\text{Area} = \frac{1}{2} \left| x_1y_2 + x_2y_3 + x_3y_1 - (y_1x_2 + y_2x_3 + y_3x_1) \right|$

– Area of $\triangle ABC$ :

$= \frac{1}{2} \left| 0 \cdot 0 + 4 \cdot 3 + 5 \cdot 0 - (0 \cdot 4 + 0 \cdot 5 + 3 \cdot 0) \right|$

$= \frac{1}{2} \left| 0 + 12 + 0 - 0 \right| = \frac{1}{2} \times 12 = 6$

– Area of $\triangle ACD$ :

$= \frac{1}{2} \left| 0 \cdot 3 + 5 \cdot 5 + 2 \cdot 0 - (0 \cdot 5 + 3 \cdot 2 + 5 \cdot 0) \right|$

$= \frac{1}{2} \left| 0 + 25 + 0 - (0 + 6 + 0) \right| = \frac{1}{2} \times 19 = 9.5$

– Area of $\triangle ADE$ :

$= \frac{1}{2} \left| 0 \cdot 3 + 2 \cdot 3 + (-1) \cdot 5 - (0 \cdot 2 + 3 \cdot (-1) + 5 \cdot 0) \right|$

$= \frac{1}{2} \left| 0 + 6 - 5 - (0 - 3 + 0) \right| = \frac{1}{2} \times 4 = 2$

3. Sum the Areas: Adding up the areas of each triangle:

$\text{Total Area} = 6 + 9.5 + 2 = 17.5 \text{ square units}$

Thus, the area of the irregular pentagon is approximately $17.5$ square units.

4. Practical Applications of Pentagon Area Calculations

Calculating the area of a pentagon has many practical applications, particularly in fields like architecture, engineering, and landscaping, where pentagonal designs might be used in structures or land areas. Knowing how to calculate both regular and irregular pentagon areas ensures accurate material estimation and spatial planning.

Examples in Real Life

Architectural Design: Some buildings or floors have pentagonal shapes, requiring

accurate area calculation for flooring, materials, and other structural components.

Landscaping: In landscaping designs, pentagonal shapes might be used in gardens, ponds, or patios. Calculating the area helps in planning layout and resource allocation.
Manufacturing and Crafts: Items like tables, tiles, and decorative pieces might have a pentagonal shape. Understanding area calculation assists in material usage and cost estimation.

Conclusion

The area of a pentagon can be calculated using straightforward formulas, especially for regular pentagons. For irregular pentagons, dividing the shape into triangles simplifies the calculation, allowing for an accurate area measurement by summing the individual triangle areas. With these methods and formulas, calculating the area of a pentagon is manageable and applicable to various real-world scenarios.