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The area of a polygon is a fundamental geometric concept, and one shape that often raises curiosity and challenges our math skills is the regular pentagon. A regular pentagon is a five-sided polygon with all sides and angles equal in measure. Calculating the area of a regular pentagon can be a tricky task as it involves the use of trigonometric formulas and a solid understanding of the properties of polygons. In this guide, we will explore the step-by-step process of calculating the area of a regular pentagon, providing you with a clear understanding of the underlying calculations and techniques required to tackle this mathematical challenge. Whether you are a student learning geometry or someone looking to freshen up your math skills, this guide will equip you with the knowledge to confidently calculate the area of a regular pentagon using various approaches. So let’s dive in and unravel the mystery of pentagon area calculations!
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A pentagon is a polygon with five straight sides. Most problems in Geometry class will revolve around a regular pentagon with five equal sides. There are two common ways to calculate the area of a regular pentagon, depending on the information the problem gives.
Steps
Find the area when the side lengths and the medians are known
- Don’t confuse a midline with a radius, which is a line connecting the center to an angle (or vertex) instead of the midpoint of the edge. If the problem only shows side lengths and radiuses, move on to the next method.
- Example 1: calculate the area of a regular pentagon with a side 3 units long and a median line 2 units long.
- In example 1, the area of the triangle = ½ x 3 x 2 = 3 units of area.
- In example 1, the area of the pentagon S = 5 x S(triangle) = 5 x 3 = 15 area units.
Find the area when the side length is known
- Example 2: calculate the area of a pentagon whose side length is 7 units.
- The base of the small triangle is ½ of the side of the pentagon. In example 2, we have the base of the small triangle = ½ x 7 = 3.5 units.
- The angle of the small triangle at the center of the pentagon is always 36º. (The center of the original pentagon is 360º, we have divided it into 10 small triangles: 360 ÷ 10 = 36. So, the angle at the center of the pentagon of each small triangle is 36º.)
- In a right triangle, the tan of an angle is equal to the length of the opposite side divided by the adjacent side.
- The side opposite the 36º angle is the base of the small triangle (½ side of the pentagon). The adjacent side of the 36º angle is the height of the small triangle.
- tan(36º) = opposite/adjacent edge
- In example 2, we have tan(36º) = 3.5 / height of the small triangle
- Height of small triangle x tan(36º) = 3.5
- Small triangle height = 3.5 / tan(36º)
- The height of the small triangle is approximately 4.8 units.
- In example 2, the area of the small triangle = ½bh = ½(3,5)(4.8) = 8.4 area units.
- In example 2, the area of the entire pentagon = 8.4 x 10 = 84 area units.
Use the formula
- Area of a regular pentagon = pa /2, where p is the circumference and a is the length of the median. ^{[2] X Research Source}
- If you don’t know the perimeter, calculate from the side lengths: p = 5s, where s is the side length.
- Area of regular pentagon = (5 s ^{2} ) / (4tan(36º)), where s is the side length.
- tan(36º) = √(5-2√5). ^{[4] X Research Source} If the calculator cannot calculate “tan”, use the formula S = (5 s ^{2} ) / (4√(5-2√5)).
- Area of regular pentagon = (5/2) r ^{2} sin(72º), where r is the radius.
Advice
- The examples in this article use rounding values to make the problem simpler. If you are measuring actual pentagons with a given side length, the results will be different.
- Irregular pentagons or pentagons with different side lengths will be more difficult to calculate the area. The most appropriate method is to divide the pentagon into triangles and calculate the area of each figure to find the sum. Depending on the case, you may need to draw a larger shape outside the pentagon, calculate the total area and then subtract the outside area.
- If possible, solve with both geometric and formulaic methods, then compare the results to double-check that your answer is correct. The two results will differ slightly (since you don’t go through the steps and round like the geometric method, but enter all the values into the formula and calculate in one go), but the difference is negligible.
- The formulas are derived from the geometric method and the article is no exception. Try to find out how to prove these formulas. Particularly, the formula for calculating the area of a pentagon from the radius will be more difficult to prove than the other formulas. Hint: you need to rely on the double angle formula.
wikiHow is a “wiki” site, which means that many of the articles here are written by multiple authors. To create this article, 25 people, some of whom are anonymous, have edited and improved the article over time.
This article has been viewed 10,261 times.
A pentagon is a polygon with five straight sides. Most problems in Geometry class will revolve around a regular pentagon with five equal sides. There are two common ways to calculate the area of a regular pentagon, depending on the information the problem gives.
In conclusion, calculating the area of a regular pentagon involves two main steps. First, find the length of the apothem by dividing the side length of the pentagon by 1.376. Then, multiply the apothem by half the perimeter of the pentagon to obtain the area. The formula for the area of a regular pentagon is A = (5s^2)/(4tan(π/5)), where A is the area and s is the side length. This formula can be used to find the area of any regular pentagon, regardless of its size. By following these steps and using the provided formula, anyone can easily calculate the area of a regular pentagon.
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