Area of a Polygon: Definitions, Formula, and Examples

The whole area covered by a polygon is called the Area of Polygon. A polygon is a closed plane figure bounded by straight line segments. It is a flat shape made up of line segments joined end to end to form a closed figure. Because a polygon can be both regular and irregular, we must use multiple methods to calculate its area depending on its shape.

Triangles, squares, rectangles, pentagons, hexagons, and other polygons are utilised in geometry. These polygons each have their own region. It is easier to find the area for polygons because their dimensions are set and known. Let’s look at some instances of how to get the area of a regular polygon.

Polygon Definition

A polygon is a closed plane figure bounded by straight line segments. Let us look at the figure below and understand the parameters required to make a polygon.

The straight-line segments that make up a polygon are called the sides of a polygon, and the endpoints of the line segments are called vertices of the polygon. We name polygons according to the number of sides they contain. For example, if the polygon has three sides, it is called a triangle. If the polygon has five sides, it is called a pentagon. If the polygon has six sides, it is called a hexagon and so on.

If each polygon angle is less than \({180^ \circ }\), the polygon is called a convex polygon, and if at least one angle of a polygon is more than \({180^ \circ }\), it is called a concave polygon. Now, as we understand what a polygon is, let us have a quick look at what an area is.

What is Area?

The area is the amount of space covered by a two-dimensional shape or surface. We measure the area of regular polygon in square units, \({\rm{c}}{{\rm{m}}^{\rm{2}}}\) or \({{\rm{m}}^{\rm{2}}}\). In the below figure, if one square block is of \({\rm{1}}\,{\rm{cm}}\) length and \({\rm{1}}\,{\rm{cm}}\) width, then we can find the area of the pink region just by counting the number of squares in the region, i.e., \({\rm{25}}\) squares.

So, the area is \(5\,{\rm{cm}} \times 5\,{\rm{cm}} = 25\,{\rm{c}}{{\rm{m}}^2}\)

Area of a Polygon Definition

The area of a polygon is the region of space occupied in a \(2 – D\) plane, irrespective of its shape, like a triangle, square, parallelogram, or trapezium. Regular polygons have definite dimensions, so it becomes easy to calculate the perimeter of a polygon when compared to other irregular polygons where the sides have no fixed dimension.

So, let us discuss how to find the area of a regular polygon and an irregular polygon.

Area of a Regular Polygon

A polygon having equal sides and equal angles are known as a regular polygon. At times with the help of apothem, we can find the area of a polygon. Apothem is a line segment that joins the centre of the polygon to the midpoint of any side, and it is perpendicular to that side. Only regular polygons have apothems. The apothem is also the radius of a circle that is drawn entirely inside the regular polygon.

The formula for the area of a regular polygon is,

\(A = \frac{{{l^2}n}}{{4\;tan\;\frac{\pi }{n}}},\) is the side length and \(n\) is the number of sides.

We can use the apothem area formula of a polygon to calculate the length of the apothem.

\(A = \frac{1}{2} \times a \times P\), where, \(A\) is the polygon area, \(a\) is the apothem, and \(P\) is the perimeter.

Area of an Irregular Polygon

An irregular polygon is a polygon with interior angles of different measures. In addition, the side lengths of an irregular polygon are also of various measurements and are included to calculate the perimeter of a polygon.

For finding the area of a polygon that is a little complex, or its formula is not defined, we split the figure into triangles, squares, trapezium, etc. The purpose is to visualize the given geometrical figure as a combination of figures we know how to calculate the area. We calculate the area for each part and then add them up to obtain the polygon area.

Like we split a quadrilateral into triangles to find its area, we can also split any polygon into triangles, trapeziums, etc., to find its area. Let us understand it with the help of an illustration.

Let us observe the above-given pentagon \(ABCDE\). Its diagonal \(AD\) is drawn. If we draw perpendicular \(BP\) and \(CQ\) on the diagonal \(AD\), then the pentagon \(ABCDE\) is divided into four parts: \(△AED, △ABP, △CQD\) then the trapezium \(BPQC.\) So, the area of the pentagon \(ABCDE\) is the sum of the areas of these four parts.

Area of a Polygon Formula

1. Triangle 

A triangle is a polygon that has three sides. It is a figure bounded or enclosed by three-line segments. 

Area of a triangle \(= \frac{1}{2} \times {\rm{base \times height}}\)

The above area of polygon formula is used when the length of any side and the corresponding height is known or given.

For the above figure, the area of the triangle \( = \frac{1}{2} \times {\rm{base \times height}} = \frac{1}{2} \times BC \times AD = \;\frac{1}{2} \times b \times h.\)

2. Square

A square is a quadrilateral, whose length is equal to its breadth. In simple words, all the sides of a square are equal.

Therefore, area of a square \({\rm{ = length \times length}}\)
\({\rm{ = side \times side}}\)
\({\rm{ = sid}}{{\rm{e}}^{\rm{2}}}\)

3. Rectangle

A rectangle is a quadrilateral whose opposite sides are equal and parallel to each other. Since a rectangle has four sides, it has four angles, each of measure \({90^ \circ }\). Let the length of the rectangle be \(l\) breadth of the rectangle be \(b\) and the area of the rectangle be \(A\), then

Area of the rectangle \( = l \times b\)

4. Parallelogram

The area of a parallelogram is the amount of space covered by it in a \(2 – D\) planar region. A parallelogram is a special type of quadrilateral. The opposite sides of a parallelogram are equal and parallel.

 The area of a parallelogram is the product of base and height.

\(A = b \times h\)

5. Rhombus

A flat shape with four equal sides and four angles which are not \({90^ \circ }\).

\({\rm{Area = }}\frac{{\rm{1}}}{{\rm{2}}}{\rm{ \times product\;of\;the\;diagonals}}\)

6. Trapezium

A trapezium has four sides with one pair of parallel sides and one pair of non-parallel sides. The area of a trapezium is,

Area of a trapezium \(\Delta ABC\; = \;\frac{1}{{2\;}} \times {\rm{base}} \times {\rm{height}} = \;\frac{1}{{2\;}} \times BC \times AM\)

Let \({b_1}\)​ and \({b_2}\)​ be the lengths of these bases. The distance between the bases is called the height of the Trapezium. Let \(h\) be the height. Then the formula becomes:

\(A = \frac{1}{2}\left( {{b_1} + {b_2}} \right) \times h\)

7. Kite

Kite is having two pairs of sides with an equal length that are adjacent to each other. The area of a kite is half the product of the lengths of its diagonals.

\({\rm{Area}} = \frac{1}{2} \times {d_1} \times {d_2}\)

8. Pentagon

A pentagon is a five-sided polygon in geometry. The five angles present in the regular pentagon are equal.

The formula calculates the area of a pentagon,

\(A = \frac{5}{2} \times s \times a\)

Where \(s\) is the side of the pentagon and \(a\) is the length of the apothem.

Area of a Polygon on a Graph

We can find the area of a polygon on a graph as well. In the example, we will calculate the area of a triangle using the coordinates.

The coordinates of the vertices of the triangle are \(A( – 1,2),B( – 3, – 1)\) and \(C(5, – 1)\)

From \(A\) draw \(AM\) perpendicular to \(BC\).

Area of \(\Delta ABC\; = \;\frac{1}{{2\;}} \times {\rm{base}} \times {\rm{height}} = \;\frac{1}{{2\;}} \times BC \times AM.\)

\( = \frac{1}{2} \times 8 \times 3 = 12\) square units.

Solved Examples

Q.1. Find the area of a regular polygon with a perimeter of \(44\,{\rm{cm}}\) and apothem length of \(10\,{\rm{cm}}.\)
Ans: As we know,
Area \((A) = \frac{1}{2} \times p \times a\), here \(p = 44\;{\rm{cm}}\) and \(a = 10\;{\rm{cm}}\)
\(= \frac{1}{2} \times 44 \times 10\;{\rm{c}}{{\rm{m}}^2}\)
\( = 220\;{\rm{c}}{{\rm{m}}^2}\)

Q.2. Find the area of a regular hexagon, each of whose sides measures \(6\,{\rm{m}}\)
Ans: For a hexagon, the number of sides \(n = 6\)
\(A = \frac{{{l^2}n}}{{4\tan \frac{\pi }{n}}}\)
By substitution, we get
\(A = \frac{{{6^2} \times 6}}{{4\tan \frac{{180}}{6}}}\)
\( = \frac{{216}}{{4\tan \frac{{180}}{6}}}\)
\( = \frac{{216}}{{2.3094}}\)
\(A = 93.53\;{{\rm{m}}^2}\)

Q.3 Calculate the area of \(5\)-sided polygon with a side length \(4\,{\rm{cm}}{\rm{.}}\)
Ans: The given parameters are,\(l = 4\;{\rm{cm}}\) and \(n = 5\)
The formula for finding the area is,
\(A = \frac{{{l^2}n}}{{4\tan \frac{\pi }{n}}}\)
By substitution, we get
\(A = \frac{{{4^2} \times 5}}{{4\tan \frac{{180}}{5}}}\)
\(A = 27.53\;{\rm{c}}{{\rm{m}}^2}\)

Q.4. In the given figure, \(AB = 8\;{\rm{m}},CE = 6\;{\rm{m}},AE = 5\;{\rm{m}}\) and \(BE = 3\,{\rm{m}}{\rm{.}}\) Find the area of the polygon \(ABCD\).

Ans: Area of \(ABCD = \;{\rm{Area\;of\;trapezium}}\;AECD\; + \;{\rm{Area\;of}}\;\Delta BCE\)
\( = \frac{1}{2} \times \left( {{b_1} + {b_2}} \right) \times h + \frac{1}{2} \times b \times h\)
\( = \frac{1}{2} \times (4 + 6) \times 5 + \frac{1}{2} \times 3 \times 6\;{{\rm{m}}^2}\)
\( = 25 + 9\;{{\rm{m}}^2}\)
\( = 34\;{{\rm{m}}^2}\)

Q.5. Find the area of a pentagon of side \(20\,{\rm{cm}}\) and apothem length \(5\,{\rm{cm}}.\)
Ans: Given
\(s = 20\;{\rm{cm}}\)
\(a = 5\;{\rm{cm}}\)
Area of a pentagon \( = A = \frac{5}{2} \times s \times a\)
\( = \frac{5}{2} \times 20 \times 5\;{\rm{c}}{{\rm{m}}^2}\)
\( = 250\;{\rm{c}}{{\rm{m}}^2}\)

Summary

In this article, we learned the basic concepts of a polygon and discovered the polygon types and how to find their areas. We also learned to calculate the area of a polygon on a graph by finding the coordinates of the polygon. We also saw how to find the area of an irregular polygon.

FAQs Based on Area of Polygon Formula

Below here we have provided some of the most asked questions related to the area of a polygon:

Q.1: What is the area of a polygon?
Ans: The area is the space occupied by a flat shape or the surface of an object. Therefore, the area of a polygon measures the size of the region enclosed by the polygon.

Q.2: How to find the area of a quadrilateral with coordinates?
Ans: When the coordinates of the vertices are given, divide the quadrilateral into two triangles by drawing the diagonal. Find the area of each triangle and then add the areas of two triangles which provides the quadrilateral area.

Q.3: What is the easiest way to calculate the area of any regular polygon?
Ans: The easiest way to calculate the area of a polygon is to divide it into triangles and apply the area of a triangle.

Q.4. What is apothem in a polygon?
Ans: The apothem of a regular polygon is a line segment from the centre to the midpoint of one of its sides.

Q.5. What is the difference between regular and irregular polygon?
Ans: Regular shapes have sides that are all equal and interior angles that are all equal. Irregular shapes have sides and angles of any length and size.