What is the area of a regular polygon?

In Geometry, a polygon is a closed two-dimensional figure made up of straight lines. We know that a triangle is a polygon with the least number of sides. Basically, the polygon can be classified as a regular or irregular polygon and convex or concave polygon. A regular polygon is a polygon in which all the sides and interior angles are equal. In this article, we are going to learn the area of a regular polygon, its formula and solved examples in detail.

Table of Contents Show

What is the Area of a Regular Polygon?
Area of Regular Polygon Formula
Area of Regular Polygon of N-sides Formula
Area of Regular Polygon Inscribed in a Circle
Area of Regular Polygon Problems and Answers
Area of Regular Polygon Practice Questions

What is the Area of a Regular Polygon?

The area of a regular polygon is the space enclosed by the boundary of the regular polygon. In other words, the area of a regular polygon is the area that is enclosed by it. It is generally measured in square units, such as cm2, m2, ft2, and so on. Generally, the area of a polygon can be determined using different formulas, based on whether the polygon is regular or irregular.

Area of Regular Polygon Formula

The area of regular polygon formulas for some of the most commonly used polygons are as follows:

Area of Equilateral triangle = (√3a2) /4 square units

Where “a” is the side length of an equilateral triangle

Area of Square = a2 square units

Where “a” is the side length of square

Area of Regular Pentagon = (1/4) ×√[5(5+2√5)] ×a2 square units

Where “a” is the side length of the pentagon

Area of Regular Hexagon = [3√3a2]/2 Square units

Where “a” is the side length of the hexagon.

Area of Regular Polygon of N-sides Formula

If “n” is the number of sides of a polygon, then the formula to find the area of regular polygon of n sides is given by:

Area of Regular Polygon Formula = [l2n]/[4tan(180/n)] Square units

Where “l” is the side length of polygon

“n” is the number of sides of the polygon.

Also, check: Area of Regular Polygon Calculator

Area of Regular Polygon Inscribed in a Circle

The area of a regular polygon inscribed in a circle formula is given by:

Area of a regular polygon inscribed in a circle = (nr2/2) sin (2π/n) square units

Where “n” is the number of sides

“r” is the circumradius.

Area of Regular Polygon Problems and Answers

Go through the below problems to find the area of a regular polygon.

Example 1:

Find the area of a regular hexagon whose side length is 2 cm.

Solution:

Given: Side length, a = 2 cm

We know that the formula for the area of a regular hexagon is [3√3a2]/2 Square units

Substituting the value in the formula, we get

A = [3√3(2)2]/2

A = (12√3)/2 = 6√3

We know that √3 = 1.732

So, A = 6(1.732) = 10.392 cm2

Therefore, the area of a regular hexagon with a side length of 2 cm is 10.392 cm2.

Example 2:

Calculate the area of a regular polygon whose side length is 6 cm and the number of sides is 5.

Solution:

Given that, the number of sides,n = 5

Side length, l = 6 cm

We know that,

Area of Regular Polygon Formula = [l2n]/[4tan(180/n)] Square units

Now, substitute the values in the formula, we get

A = [(62(5)]/[4tan(180/5)] cm2

A = 180/[4tan (180/5)] cm2

A = 180/ [4(tan 36)] cm2

A = 180/[4(0.7265)] cm2

A = 180/2.906 cm2

A = 61.94 cm2

Hence, the area of a regular polygon is 61.94 cm2.

Example 3:

Find the area of regular pentagon inscribed in a circle whose circumradius is 4 cm.

Solution:

Given: Number of sides, n = 5

Circumradius, r = 4 cm.

We know that the area of a regular polygon inscribed in a circle = (nr2/2) sin (2π/n) square units.

Substituting the values, we get

A = (5(4)2/2) sin (2π/5) cm2

A = (5(16)/2) sin (360/5) cm2

A = 40 sin (72) cm2

A = 40(0.951) cm2

A = 38.04 cm2

Therefore, the area of a regular pentagon inscribed in a circle is 38.04 cm2.

Area of Regular Polygon Practice Questions

Solve the following problems:

Find the area of a regular hexagon whose side length is 4 cm.
Compute the area of a regular polygon of side length 6 cm, whose number of sides is 7.
Determine the area of a regular pentagon whose side length is 8 cm.
Find the area of a regular pentagon inscribed in a circle whose circumradius is 4 cm.

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The area of a regular polygon is the space enclosed by the boundary of the regular polygon. The area of a regular polygon is generally measured in square units.

The formula for the area of a regular polygon of n-sides is [l2n]/[4tan(180/n)] Square units, where l is the side length of the polygon and n is the number of sides.

The area of a regular polygon inscribed in a circle formula is (nr2/2) sin (2π/n) square units, where n is the number of sides and r is the circumradius.

No, the area of a regular polygon does not directly depend on the exterior angle.

The area of a regular polygon is the space occupied by it, whereas the perimeter of a regular polygon is the total length of the boundary of a regular polygon.

A regular polygon, remember, is a polygon whose sides and interior angles are all congruent. To understand the formula for the area of such a polygon, some new vocabulary is necessary.

The center of a regular polygon is the point from which all the vertices are equidistant. The radius of a regular polygon is a segment with one endpoint at the center and the other endpoint at one of the vertices. Thus, there are n radii in an n-sided regular polygon. The center and radius of a regular polygon are the same as the center and radius of a circle circumscribed about that regular polygon.

An apothem of a regular polygon is a segment with one endpoint at the center and the other endpoint at the midpoint of one of the sides. The apothem of a regular polygon is the perpendicular bisector of whichever side on which it has its endpoint. A central angle of a regular polygon is an angle whose vertex is the center and whose rays, or sides, contain the endpoints of a side of the regular polygon. Thus, an n-sided regular polygon has n apothems and n central angles, each of whose measure is 360/n degrees. Every apothem is the angle bisector of the central angle that contains the side to which the apothem extends. Below are pictured these characteristics of a regular polygon.

Figure %: A regular polygon with a center (C), radius (r), apothem (a), and central angle

Once you have mastered these new definitions, the formula for the area of a regular polygon is an easy one. The area of a regular polygon is one-half the product of its apothem and its perimeter. Often the formula is written like this: Area=1/2(ap), where a denotes the length of an apothem, and p denotes the perimeter.

When an n-sided polygon is split up into n triangles, its area is equal to the sum of the areas of the triangles. Can you see how 1/2(ap) is equal to the sum of the areas of the triangles that make up a regular polygon? The apothem is equal to the altitude, and the perimeter is equal to the sum of the bases. So 1/2(ap) is only a slightly simpler way to express the sum of the areas of the n triangles that make up an n-sided regular polygon.

Figure %: Two n-sided polygons divided into n triangles

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This area of a regular polygon calculator can help - as you can guess - in determining the area of a regular polygon. Type the number of sides and the polygon area appears in no time. If you're wondering how to find the area of a polygon formula, keep reading and you'll find the answer! If you want to calculate the area of any 3-sided or 4-sided polygon, check out this triangle area calculator and quadrilateral area calculator.

The most popular, and usually the most useful formula is the one that uses the number of sides $n$ and the side length $a$ :

$\times a^2 \times \frac{1}{4}\cot\left(\frac{\pi}{n}\right)$

However, given other parameters, you can also find out the area:

$\times a \times r_\text{i} / 2$ , having $rir_\text{i}$ - incircle radius (it's also an apothem - a line segment from the center to the midpoint of one of its sides).
$\text{perimeter}\times r_\text{i} / 2$ , given $rir_\text{i}$ and the polygon perimeter.
$\times r_\text{i}^2 \times \tan{(\pi/n)}$ , given $rir_\text{i}$ .
$\times r_\text{c}^2 \times \sin{(2\pi/n)} / 2$ , having $rcr_\text{c}$ - circumcircle radius.

🙋 The circumcircle is the circle that passes through all the polygon's vertex: you can learn how to calculate its center in the case of a triangle at our circumcenter of a triangle calculator.

If you want to find the area of a regular polygon, simply use the formulas described above. However, if the polygon is not regular (what means it isn't equiangular and equilateral), you can:

Find the area using vertices coordinates:

$⁣xi+1)∣2A\! =\! \frac{\left|\sum (x_i \!\times\! y_{i+1})\! -\! (y_i \!\times\! x_{i+1})\right|}{2}$

with $\rightarrow x(1)$ and $\rightarrow y(1)$

Determine the polygon area given side lengths and some diagonals, by splitting the polygon into triangles. Then find the area with given three sides (SSS) equation (you can learn the origin of this formula with our Heron's formula calculator).
Calculate the area of polygons using other formulas - e.g. for a scalene triangle or a quadrilateral.

Let's assume that you want to calculate the area of a specific regular polygon, e.g. 12-sided polygon, dodecagon with 5-inch sides.

Enter the number of sides of chosen polygon. Put $12$ into the number of sides box.
Type in the polygon side length. In our example, it's equal to $in5\ \text{in}$ .
Our area of polygon calculator displays the area. It's $in2279.9\ \text{in}^2$ .

If you want to calculate the area of a regular polygon using other parameters than the side length, check out this general regular polygon calculator.

To calculate the area of a regular polygon given the radius apply the formula:
area = n * a² * cot(π/n) / 4
Where:

n is the number of sides of the polygon;
a is the length of the side; and
cot is the cotangent function (cot(x) = 1/tan(x)).

15.484. To compute the area of a pentagon with side 3, you can directly apply the formula:
area = n × a² × cot(π/n) / 4 ,
substituting:

If you know the apothem of a regular polygon you can compute the area with the formula:
area = ap² × n × tan(π/n)
Where:

ap is the apothem (the segment connecting the side midpoint to the center); and
n is the number of sides of the regular polygon.

1.509 m². To calculate the area, use the formula for the area of a regular hexagon:
area = 6 × a² × cot(π/6) / 4
Where a is the side length.
This particular hexagon is equal in size to any one of the 18 elements in the primary mirror of the James Webb Space Telescope.