How to Calculate the Area of a Regular Pentagon

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.

Table of Contents

Find the area when the side lengths and the medians are known

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Start with side lengths and midlines. This method applies to regular pentagons with five equal sides. In addition to the side length, you need to know the “midline” measure. The midline of a regular pentagon is a perpendicular segment from the center to one side.

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.

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Divide the pentagon into five triangles. Draw five lines from the center to each vertex. You will have five triangles.

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Calculate the area of the triangle. Each triangle has the base as the side of the pentagon and the height as the midline. (Remember: the height of a triangle is the line that goes from the vertex to the opposite side and forms a right angle.) To find the area of a triangle, you simply calculate ½ x base x height.

In example 1, the area of the triangle = ½ x 3 x 2 = 3 units of area.

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Multiply the area of a triangle by 5. Since we’ve divided the pentagon into five equal triangles, to calculate the total area, you simply multiply the area of one triangle by 5.

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

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Start with just the side lengths. This method applies to regular pentagons with five equal sides.

Example 2: calculate the area of a pentagon whose side length is 7 units.

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Divide the pentagon into five triangles. Draw a line from the center of the pentagon to each vertex. You need to draw a total of five lines. At this point, we will have five triangles of the same size.

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Divide the triangle in half. Draw a line from the center of the pentagon to the base of the triangle. This line will be perpendicular to the base and divide the triangle into two smaller triangles of the same size.

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Find the area of the small triangle. Start calculating the side lengths and angles of the small triangle as follows:

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.)

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Find the height of the small triangle. The height of the small triangle is the line segment joining the perpendicular from the center to the side of the pentagon. We can use the trigonometric formula to calculate this side length: ^{[1] X Research Source}

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.

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Calculate the area of the triangle . The area of the triangle is ½ base x height. Formula: S = ½bh, where b is the length of the base and h is the height of the triangle. Now that you know the height and length of the base of the small triangle, plug the values into the formula to find the area.

In example 2, the area of the small triangle = ½bh = ½(3,5)(4.8) = 8.4 area units.

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Do the multiplication to find the area of the pentagon. A small triangle is 1/10 the area of the pentagon. So, to calculate the area of a pentagon, you just need to multiply the area of the small triangle by 10.

In example 2, the area of the entire pentagon = 8.4 x 10 = 84 area units.

Use the formula

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Use perimeter and midline. The midline of a regular pentagon is a perpendicular segment from the center to one side. If the problem gives the length of the midline, you can easily calculate the area of the pentagon using the formula below:

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.

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Use side lengths. If you only know the side lengths, apply the following formula: ^{[3] X Research Source}

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)).

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Select a formula that uses only the radius. If the problem is only for radius, you can apply this formula: ^{[5] X Research Source}

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.

Steps

Find the area when the side lengths and the medians are known

Find the area when the side length is known

Use the formula

Advice

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