Area of a Pentagon Formula: In geometry, we study different shapes. The \(2\)-dimensional shape made up of only straight line segments is known as a polygon. A pentagon is a simple five-sided polygon. The sum of all the internal angles of a polygon is equal to \({540^ \circ }.\) The name pentagon was taken from the Greek word Penta and Gonia. Penta means five, and Gonia means angles.

In this article, we will learn in detail about the definition of the pentagon, properties of a pentagon, different types of pentagons, and formulas to calculate the area and perimeter of a regular pentagon.

What is a Pentagon?

A pentagon is a five-sided polygon with five straight lines and five interior angles, which add up to \({540^ \circ }.\) A simple pentagon required five straight sides that meet to create five vertices but do not bisect with each other.

The meaning of pentagon shape is derived from the Greek word as Penta denotes five, and gonia denote angle. If we trace the boundary of a cupcake that has icing on its top, we can easily imagine a pentagon shape.

Types of Pentagon

Any polygon has four different types: concave polygon, convex polygon, regular polygon, and irregular polygon. Similarly, the pentagon has four types. They are,

1. Concave Pentagon
2. Convex Pentagon
3. Regular Pentagon
4. Irregular Pentagon

Concave Pentagon

A pentagon in which at least one angle is more than \({180^ \circ }\) is called a concave pentagon. In other words, if the vertices point inwards or pointing inside a pentagon, it is known as a concave pentagon.

Here, in the pentagon \(ABCDE,\) we can see that the interior angle \(\angle ABC\) is more than \({180^ \circ }.\) Hence, this is a concave pentagon.

Convex Pentagon

A pentagon in which each angle is less than \({180^ \circ }.\) is called a convex pentagon. In other words, if the vertices point outwards in a pentagon or pointing outside is known as a convex pentagon.

Regular Pentagon

A polygon is said to be regular if all the sides and angles of it are equal. A pentagon with all sides equal and all the angles equal is called a regular pentagon.

The pentagon \(ABCDE\) is a regular pentagon if \(AB = BC = CD = DE = EA\) and \(\angle A = \angle B = \angle C = \angle D = \angle E.\)

Irregular Pentagon

Pentagons that aren’t regular are called irregular pentagons. In other words, a pentagon in which the sides and angles have different measures is known as an irregular pentagon.

Interior and Exterior Angle of a Regular Pentagon

We know that, for a regular polygon of \(n\) sides, we have
Sum of exterior angles equal to \({360^ \circ }.\)
Each exterior angle \(= \frac{{{{360}^ \circ }}}{n}\)
Sum of interior angles of a polygon\( = \left({n – 2} \right) \times {180^ \circ }\)
Each interior angle \(= {180^ \circ } – \)(each exterior angle)
\( = {180^ \circ } – \frac{{{{360}^ \circ }}}{n}\)
\( = \frac{{n \times {{180}^ \circ } – 2\left({2 \times {{180}^ \circ }} \right)}}{n}\)
\( = \frac{{\left({n – 2} \right) \times{{180}^ \circ }}}{n}\)
Therefore, interior angle \( = \frac{{\left({n – 2} \right) \times{{180}^ \circ }}}{n}\)
So, the sum of interior angles of a pentagon\( = \left({n – 2} \right) \times {180^ \circ }\)
\( = \left({5 – 2} \right) \times {180^ \circ }\)
\( = 3 \times {180^ \circ }\)
\( = {540^ \circ }\)
The measure of each interior angle of a regular polygon \( = \frac{{\left({5 – 2} \right) \times {{180}^ \circ }}}{5} = {108^ \circ }\)
The measure of each exterior angle of a regular pentagon \( = \frac{{{{360}^ \circ }}}{5} = {72^ \circ }\)

Perimeter of a Pentagon

The computation of the length of the boundary of any closed figure is known as its perimeter. The perimeter of a regular or irregular pentagon will be the sum of the lengths of its sides.

If \(a,b,c,d,e\) are respectively the lengths of sides of an irregular pentagon, then the perimeter of an irregular pentagon is \(a + b + c + d + e.\)
That is, perimeter \(p = a + b + c + d + e\)
If a represents the side of a regular pentagon, then the perimeter of a regular pentagon \( = 5a.\)
That is, perimeter \(P = 5a\)

Apothem

A line drawn from the centre of any polygon to the mid-point of one of the sides is known as apothem.

In the above hexagon, \(a\) is the apothem.
The formula to find the apothem is based on the side length and the radius.
When the side length is given, then the apothem formula is given by
\(a = \frac{S}{{2\tan \left({\frac{{180}}{n}} \right)}}\)
Where \(a = \) apothem length
\(s = \) length of the side of the pentagon
\(n = \) number of sides in the given polygon
When the radius is given, the apothem formula is given by
\(a = r\cos \frac{{180}}{n}\)
Where \(r = \) radius
\(n = \) number of sides in the given polygon

Area of Regular Pentagon with Apothem

Assume a pentagon is having side \(s\) and length of apothem \(a.\) The area of the pentagon is equal to \(\frac{5}{2}\) times the length of the apothem and the side of the pentagon.
The formula to calculate the area of the pentagon is \(\frac{5}{2} \times s \times a\,{\text{sq}}{\text{.units}}\)

Area of a Regular Pentagon without Apothem

The area of a regular pentagon can similarly be expressed in terms of length of side \(a.\)
The area of a regular pentagon with side length a is given by
\(A = \frac{1}{4}\sqrt {5\left({5 + 2\sqrt 5 } \right)} {a^2}\)
Also, we can use another formula to find the area of the pentagon when only the side length is given. It is given by
\(A = \frac{{5{a^2}}}{{4\,\tan \,{{36}^ \circ }}}\)

Area of Regular Pentagon with Radius

Let \(r\) be the length of the radius of a regular pentagon. Then the area of a regular pentagon is given by
\(A = \frac{5}{2}{r^2}\sin \,{72^ \circ }\)

Solved Example – Area of a Pentagon Formula

Q.1. Find the area of the given regular pentagon whose side measure is \(4\,{\text{cm}}.\)
Ans: We know that the area of a regular pentagon with side measure a units is given by
\(A = \frac{{5{a^2}}}{{4\,\tan \,{{36}^ \circ }}}\)
So, the area of a regular pentagon with a side measure of \(4\,{\text{cm}}\) is \(A = \frac{{5{{\left( 4 \right)}^2}}}{{4\,\tan \,{{36}^ \circ }}}\)
\( = \frac{{5 \times 16}}{{4 \times 0.726542528}}~{\text{c}}{{\text{m}}^2}\)
\( = \frac{{80}}{{2.906170112}}~{\text{c}}{{\text{m}}^2}\)
\( = 5.50552~{\text{c}}{{\text{m}}^2}\)
Therefore, the area of the regular polygon with a side measure of \(4\,{\text{cm}}\) is \(5.50552\,{\text{c}}{{\text{m}}^2}\)

Q.2. Find the area of the given regular pentagon whose side measure is \(3\,{\text{cm}}.\)
Ans: We know that the area of a regular pentagon with side measure \(a\) units is given by
\(A = \frac{1}{4}\sqrt {5\left({5 + 2\sqrt 5 } \right)} {a^2}\)
Therefore, \(A = \frac{1}{4}\sqrt {5\left({5 + 2\sqrt 5 } \right)} \times {3^2}\)
\( = \frac{1}{4}\sqrt {5\left({5 + 2\sqrt 5 }\right)} \times 9\)
\( = 15.484~{\text{c}}{{\text{m}}^2}\)
Therefore, the area of a regular pentagon with a side of \(3\,{\rm{cm}}\) is \(15.484\,{\rm{c}}{{\rm{m}}^2}.\)

Q.3. Find the area and perimeter of a regular pentagon whose side is \(6\,{\text{cm,}}\) and apothem length is \(5\,{\text{cm.}}\)
Ans: Given: side measure of regular pentagon \(s = 6\,{\text{cm}}\)
The measure of apothem \(a = 5\,{\text{cm}}\)
We know that, area of a regular pentagon \(A = \frac{5}{2} \times s \times a\,{\text{sq}}{\text{.units}}\)
\( \Rightarrow A = \frac{5}{2} \times 6 \times 5~{\text{c}}{{\text{m}}^2}\)
\( \Rightarrow A = 75~{\text{c}}{{\text{m}}^2}\)
Now, we know that the perimeter of a regular pentagon with side length \(a\) units is \(5a.\)
Therefore, the perimeter of the given pentagon \( = 5 \times 6\,{\text{cm}}\)
\( = 30\,{\text{cm}}\)
Therefore, the area of the given pentagon is \(75\,{\text{c}}{{\text{m}}^2}\) and the perimeter is \({\text{30}}\,{\text{cm}}{\text{.}}\)

Q.4. Find the pentagon area whose length of the side is \(16\,{\text{units}}\) and the length of apothem is \(5\,{\text{units}}.\)
Ans: Given side measure of regular pentagon \(s = 16\,{\text{units}}\)
The measure of apothem \(a = 5\,{\text{units}}\)
We know that, area of a regular pentagon \(A = \frac{5}{2} \times s \times a\,{\text{sq}}{\text{.units}}\)
\( = \frac{5}{2} \times 16 \times 5\)
\({\text{=200 sq}}{\text{. units}}\)
Therefore, the area of the given pentagon is \({\text{200 sq}}{\text{. units}}{\text{.}}\)

Q.5. Raghu was given the area of a pentagon as \({\text{300 units}}\) square and having a side of \({\text{15 units}}.\) Can you help him find the length of the apothem of the pentagon?
Ans: Given side measure of regular pentagon \(s = 15\,{\text{units}}\)
Area of the pentagon \( = 300\,{\text{sq}}.\,{\text{units}}\)
We know that, area of a regular pentagon with side \(s\) and the measure of apothem \(a\) is found by
\(A = \frac{5}{2} \times s \times a\,{\text{sq}}{\text{.units}}\)
\( \Rightarrow 300 = \frac{5}{2} \times 15 \times a\,{\text{sq}}{\text{.units}}\)
\( \Rightarrow a = \frac{{300 \times 2}}{{5 \times 15}}{\text{units}}\)
\( \Rightarrow a = 8\,{\text{units}}{\text{.}}\)
Therefore, the length of the apothem is \(8\,{\text{units}}{\text{.}}\)

Summary

In this article, we have learned the definition of pentagon, different types of a pentagon, formula to find the area of the pentagon with apothem, formula to find the area without apothem, formula to find the area of pentagon when the radius is given, and the formula to find the perimeter of a pentagon. Also, we have solved some example problems based on the formula of area and perimeter of the pentagon.

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Frequently Asked Questions (FAQs) – Area of a Pentagon

Frequently asked questions related to area of a pentagon is listed as follows:

Q.1. How do you find the area of a pentagon formula?
Ans: The area of a regular pentagon with \(a\) as the length of the side is given by \(A = \frac{1}{4}\sqrt {5\left({5 + 2\sqrt 5 } \right){a^2}} .\)

Q.2. What is the formula to find the pentagon area when the apothem length is known?
Ans: The formula to find the area of pentagon when the length of the apothem \(a\) and side length \(s\) is known is given by
\(A = \frac{5}{2} \times s \times a\,{\text{sq}}{\text{.units}}\)

Q.3. What is the formula to find the pentagon area when the apothem length is unknown?
Ans: We have two formulas to calculate the area of the pentagon without an apothem. They are
\(A = \frac{1}{4}\sqrt {5\left({5 + 2\sqrt 5 } \right)}{a^2}\) and \(A = \frac{{5{a^2}}}{{4\,\tan \,{{36}^ \circ }}}\)
Where \(a\) is the length of the side of the pentagon.

Q.4. What is the formula to find the pentagon area when the radius length is known?
Ans: Let \(r\) be the length of the radius of a regular pentagon. Then the area of a regular pentagon is given by
\(A = \frac{5}{2}{r^2}\sin {72^ \circ }\)

Q.5. What is the interior and exterior angles of a regular pentagon?
Ans: The sum of exterior angles of a regular pentagon is \({360^ \circ }.\)
Each exterior angle \( = \frac{{{{360}^ \circ }}}{n} = \frac{{{{360}^ \circ }}}{5} = {72^ \circ }\)
The sum of interior angles of a regular pentagon is \(\left({n – 2} \right) \times {180^ \circ } = {540^ \circ }\)
Therefore, the measure of each interior angle \( = \frac{{\left({n – 2} \right) \times{{180}^ \circ }}}{n} = {108^ \circ }\)

We hope this detailed article on the area of a pentagon formula helped you in your studies. If you have any doubts, queries or suggestions regarding this article, feel to ask us in the comment section and we will be more than happy to assist you. Happy learning!