Area of a Polygon

• Introduction
• Understanding the Area of Polygon
• Difference Between Perimeter and Area of Polygons
• Area of Polygon Formulas
• Calculating Area of Regular Polygons
• Calculating Area of Irregular Polygons
• Calculating Areas of Polygons with Coordinates
• Solved Examples
• Practice Problems
• Frequently Asked Questions

Introduction

The area of a polygon is basically the space inside the boundary lines of a polygon. Imagine drawing a shape on a piece of paper and then coloring it in; that colored area is its area. A polygon is any shape made by connecting straight lines to form a closed figure, like a triangle or a square. Each polygon has a certain number of sides, and depending on the type of polygon, there are different ways to figure out its area. For instance, if you have a square, you can find its area by multiplying the length of one side by itself. Similarly, other polygons have their own unique formulas for finding area. 

In this lesson, we'll see how to find the area of different types of polygons, whether they're regular or irregular, and we'll go through some examples to help grasp the concept better.

Understanding the Area of Polygon

The area of a polygon is simply the space it takes up in a flat, two-dimensional plane. Picture it like this: if you draw a shape on a piece of paper, the area is the amount of that paper the shape covers. We measure this space in square units, like square meters or square centimeters. 

Polygons can be of two types: regular and irregular. 

Regular polygons: A polygon having all their sides and angles equal, like a square or an equilateral triangle is known as a regular polygon. Calculating the area of regular polygons is often easier, as we can use specific formulas tailored to each shape.

Irregular polygons: A polygon having sides and angles that aren't all the same, like a rectangle is called an irregular polygon. However, for irregular polygons, we may need to break them down into simpler shapes, find the area of each piece, and then add them together to get the total area. 

Difference Between Perimeter and Area of Polygons

To understand perimeter and area of polygons, we need to grasp their basic differences. A polygon is a closed shape formed by joining straight lines. Its perimeter is the total length around its boundary, while its area is the space inside the boundary. Let's look at a table to see the contrast between them.

Area of Polygon Formulas

A polygon can be classified as either a regular or an irregular polygon based on the lengths of its sides. Before calculating the area of a polygon, it's essential to determine whether the given polygon is a regular or an irregular polygon. This distinction affects how we calculate the polygon's area. 

The formulas for the areas of some commonly known polygons are:

• Area of a triangle: `\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}`
• Area of a rectangle: \( \text{Area} = \text{length} \times \text{width} \)
• Area of a parallelogram: \( \text{Area} = \text{base} \times \text{height} \)
• Area of a trapezium: `\text{Area} = \frac{1}{2} \times \left(\text{sum of lengths of its parallel sides or bases}\right) \times \text{height}`
• Area of a rhombus: `\text{Area} = \frac{1}{2} \times \text{product of diagonals}`

Example: Find the area of a trapezoid with bases \( b_1 = 6 \) units, \( b_2 = 10 \) units, and height \( h = 5 \) units.

Solution: 

Given: 

\( b_1 = 6 \)

\( b_2 = 10 \)

\( h = 5 \) 

Area of the trapezoid can be found using the formula:

`\text{Area} = \frac{1}{2} \times (b_1 + b_2) \times h`

`= \frac{1}{2} \times (6 + 10) \times 5`

`= \frac{1}{2} \times 16 \times 5`

\( = 40 \text{ square units} \)

Hence, the area of the trapezoid is \( 40 \text{ square units} \).

Calculating the Area of Regular Polygons

A shape with equal sides and equal angles is termed a regular polygon. To find the area of these polygons, we use specific formulas for each type. Let's check out some of the common formulas for finding the area of regular polygons.

• Area of Equilateral Triangle ` = \frac{\sqrt{3} \times (\text{length of a side})^2}{4}`
• Area of Square \( = (\text{length})^2 \)
• Area of Regular Pentagon `= \frac{5}{2} \times \text{side length} \times \text{length of the apothem}`
• Area of Regular Hexagon `= \frac{3\sqrt{3} \times (\text{length of a side})^2}{2}`

However, the area of a regular polygon is also calculated using the generic formula:

`\text{Area of Regular Polygon} = \frac{1}{2} \times \text{Perimeter} \times \text{Apothem}`

This formula can also be expressed as:

`\text{Area of Regular Polygon} = \frac{1}{2} \times (\text{Number of Sides}) \times (\text{Length of One Side}) \times (\text{Apothem})`

where the length of apothem is given as `\frac{l}{2 \tan\left(\frac{180}{n}\right)}`.

Calculating the Area of Irregular Polygons

An irregular polygon is a shape with unequal sides and angles. To find its area, we break it into smaller, known shapes like triangles or quadrilaterals. Then, we calculate the area of each shape separately and finally add them to determine the total area of the figure.

\( \text{Area of Quadrilateral } (ABCD) = \text{(Area of Triangle } ABD) + \text{(Area of Triangle } BCD) \)

In the above figure, the diagonal \( BD \) will act as a common base of the two triangles \( ABD \) and \( BCD \) with heights \( a \) and \( b \) respectively.

`\text{Area of Quadrilateral } (ABCD) = \text{Area of Triangle } (ABD) + \text{Area of Triangle } (BCD)`

`\text{Area of Quadrilateral } (ABCD) = \frac{1}{2} \times BD \times a + \frac{1}{2} \times BD \times b`

`\text{Area of Quadrilateral } (ABCD) = \frac{1}{2} \times BD \times (a + b)`

Calculating Areas of Polygons with Coordinates

The area of polygons with coordinates can be calculated using the following steps:

Step `1`: Compute the distance between each pair of points using the distance formula, \( D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \).

This provides us with the dimensions of the polygons.

Step `2`: Assess whether the polygon is regular or irregular.

Step `3`: For regular polygons, we use the formula, 
`\text{Area of Regular Polygon} = \frac{\text{Number of Sides} \times \text{Length of One Side} \times \text{Apothem}}{2}`

where the length of apothem is given as ` \frac{l}{2 \tan\left(\frac{180}{n}\right)} `.

`l` is the length of one side

`n` is the number of sides

Step `4`: For irregular polygons, we break them down into smaller regular polygons, calculate the area of each, and sum them to find the total area of the irregular shape.

Let’s illustrate this with an example.

Example: Find the area of the polygon with coordinates \( A = (0, 0) \), \( B = (0, 4) \), \(C = (4, 4) \) and \( D = (4, 0) \). 

Solution:  Using the distance formula \( D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \), we find the lengths of the sides:

`AB = \sqrt{(0 - 0)^2 + (4 - 0)^2} = \sqrt{0^2 + 4^2} = 4`

`BC = \sqrt{(4 - 0)^2 + (4 - 4)^2} = \sqrt{4^2 + 0^2} = 4`

`CD = \sqrt{(4 - 4)^2 + (0 - 4)^2} = \sqrt{0^2 + (-4)^2} = 4`

`DA = \sqrt{(0 - 4)^2 + (0 - 0)^2} = \sqrt{(-4)^2 + 0^2} = 4`

Since all sides are of equal length, it's a regular polygon. We can use the formula for polygon area for a regular polygon.

The length of one side, \( a \), is `4`. 

Length of the apothem  `=\frac{a}{2 \tan\left(\frac{180}{4}\right)}  = \frac{4}{2 \tan\left(45\right)} = \frac{4}{2}`

For a square, the apothem is equal to half the side length, which is `a/2 = 4/2 = 2`.

Using the formula for polygon area, `\text{Area of Polygon} = \frac{\text{Number of Sides} \times \text{Length of One Side} \times \text{Apothem}}{2}`, we plug in the values:

`\text{Area} = \frac{4 \times 4 \times 2}{2} = \frac{32}{2} = 16`

Therefore, the area of the square is `16` square units.

Solved Examples

Example `1`: Find the area of a regular hexagon that has a side length of \( s = 6 \) units.

Solution: 

Side length \( s = 6 \) units

For a regular hexagon

`\text{Area} = \frac{3\sqrt{3} \times (\text{length of a side})^2}{2}`

`= \frac{3\sqrt{3} \times (6)^2}{2}`

`= \frac{3\sqrt{3} \times 36}{2}`

`= 3\sqrt{3} \times 18`

`= 54\sqrt{3} \text{ square units}`

Therefore, the area of a regular hexagon is \( 54\sqrt{3} \text{ square units} \).

Example `2`: Find the area of an equilateral triangle of side \( 10 \) cm.

Solution:  

Side length, \( s = 10\, \text{cm} \)

For an equilateral triangle, area (\( A \)) can be calculated as:

`\text{Area} = \frac{\sqrt{3} \times (\text{length of a side})^2}{4}`

`= \frac{\sqrt{3} \times (10)^2}{4}`

`= \frac{\sqrt{3} \times 100}{4}`

`= 25 \sqrt{3}\ \text{cm}^{2}`

So, the area of the equilateral triangle is \( 25 \sqrt{3}\, \text{cm}^{2} \).

Example `3`: Find the area of the given polygon `PQRSTU`.

Solution: 

It is evident that the given polygon is irregular. To find its area, we divide the polygon into a square `(PQRU)` and a trapezium `(RSTU)`. Let's list the given values:

For square `PQRU`:

Length of PQ \( = 12\, \text{cm} \)

Thus, the area of square `PQRU` is:

`\text{Area of Square PQRU} = (\text{PQ})^2`

`= (12)^2`

`= 144\ \text{cm}^{2}`

For trapezium `RSTU`:

Length of ST \( = 8\, \text{cm} \)

Length of RU \( = 12\, \text{cm} \)

Height of the trapezium \( = 5\, \text{cm} \)

Hence, the area of trapezium `RSTU` is calculated as:

`\text{Area of trapezium RSTU} = \frac{1}{2} \times (\text{sum of lengths of bases}) \times \text{height}`

`= \frac{1}{2} \times (8 + 12) \times 5`

`= \frac{1}{2} \times 20 \times 5`

`= 50\ \text{cm}^{2}`

The total area of the polygon `PQRSTU`: 

`\text{Area of polygon PQRSTU} = \text{Area of Square PQRU} + \text{Area of trapezium RSTU}`

`= 144\ \text{cm}^{2} + 50\ \text{cm}^{2}`

`= 194\ \text{cm}^{2}`

Therefore, the area of the given polygon is \(194 \, \text{cm}^{2} \)

Example `4`: Find the area of the polygon with coordinates \( P = (1, 1) \), \( Q = (5, 1) \), \(R = (5, 4) \) and \( S = (1, 4) \). 

Solution:  

Using the distance formula \( D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \), we find the lengths of the sides:

`PQ = \sqrt{(5 - 1)^2 + (1 - 1)^2} = \sqrt{4^2 + 0^2} = 4`

`QR = \sqrt{(5 - 5)^2 + (4 - 1)^2} = \sqrt{0^2 + 3^2} = 3`

`RS = \sqrt{(1 - 5)^2 + (4 - 4)^2} = \sqrt{(-4)^2 + 0^2} = 4`

`SP = \sqrt{(1 - 1)^2 + (1 - 4)^2} = \sqrt{0^2 + (-3)^2} = 3`

We see that `PQ = RS` and `QR = SP`. Hence, `PQRS` is a rectangle.

The length of one side, \( l \), is `4` and the length of the other side, `b`, is `3`. For a rectangle:

`\text{Area} = \text{l} \times \text{b}`

`= 4 \times 3`

`= 12\ \text{cm}^{2}`

So, the area of the rectangle with vertices \( P(1, 1) \), \( Q(5, 1) \), \( R(5, 4) \), and \( S(1, 4) \) is \( 12\, \text{cm}^{2} \).

Example `5`: Find the area of a regular pentagon with a side length of \(12\) cm and an apothem of \(9\) cm.

Solution: 

Side length, \( s = 12\, \text{cm} \)

Apothem, \( a = 9\, \text{cm} \)

As the polygon is a pentagon having five sides, so,

Perimeter (\(p\)) is \(= (5 \times s) = (5 \times 12) = 60\) cm.

Now, as we know,

`\text{Area} (A) = \frac{1}{2} \times p \times a`

`= \frac{1}{2} \times 60 \times 9`

`= 270 \ \text{cm}^2`

Thus, the area of the regular polygon is \( 270 \, \text{cm}^2 \).

Practice Problems

Q`1`.  Find the area of the polygon with coordinates \( P = (0, 0) \), \( Q = (0, 2) \), \(R = (2, 2) \) and \( S = (2, 0) \). 

1. \( 2\, \text{square units} \)
2. \( 4\, \text{square units} \)
3. \( 6\, \text{square units} \)
4. \( 8\, \text{square units} \)

Answer: b

Q`2`. Find the area of a regular polygon with perimeter of `44` cm and apothem length of `10` cm.

1. \( 175\, \text{cm}^2 \)
2. \( 198\, \text{cm}^2 \)
3. \( 220\, \text{cm}^2 \)
4. \( 260\, \text{cm}^2 \)

Answer: c

Q`3`.  Find the area of the given polygon `ABCDEF`.

1. \( 545\, \text{cm}^2 \)
2. \( 630\, \text{cm}^2 \)
3. \( 710\, \text{cm}^2 \)
4. \( 789\, \text{cm}^2 \)

Answer: b

Q`4`. Find the area of a regular hexagon that has a side length of \( 18 \) units.

1. \( 184\sqrt{3} \text{ square units} \)
2. \( 290\sqrt{3} \text{ square units} \)
3. \( 358\sqrt{3} \text{ square units} \)
4. \( 486\sqrt{3} \text{ square units} \)

Answer: d

Q`5`. Find the area of the given polygon

1. \( 122\, \text{in}^2 \)
2. \( 132\, \text{in}^2 \)
3. \( 136\, \text{in}^2 \)
4. \( 144\, \text{in}^2 \)

Answer: a

Frequently Asked Questions

Q1. What is the formula for finding the area of a polygon?

Answer: The formula for finding the area of a polygon depends on the type of polygon. For a regular polygon with \(n\) sides, each of length \(a\), and apothem \(A\), the formula for polygon area is:

` \text{Area of Polygon} = \frac{\text{Number of Sides} \times \text{Length of One Side} \times \text{Apothem}}{2}`

Q`2`. How to find the area of a polygon that is irregular?

Answer: To find the area of an irregular polygon, you can divide it into smaller shapes (triangles, rectangles, etc.) whose areas you can calculate using known formulas. Then, sum up the areas of these smaller shapes to get the total area of the irregular polygon.

Q`3`. What is the apothem of a polygon?

Answer: The apothem of a polygon is the perpendicular distance from the center of the polygon to one of its sides. In a regular polygon, all apothems have the same length.

Q`4`. How do you find the area of a triangle?

Answer: The area \(A\) of a triangle with base \(b\) and height \(h\) can be calculated using the formula:

`A = \frac{1}{2} \times b \times h`

Q`5`. Can you find the area of a polygon without knowing the coordinates of its vertices?

Answer: Yes, you can find the area of a polygon without knowing its coordinates if you have information about its side lengths, angles, or other properties that allow you to calculate the lengths of its sides and apothem.