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The area of a polygon is referred to as the area occupied by a closed polygon. Polygons can be regular or irregular.
- The basic polygons include the triangle, square, rectangle, pentagon, hexagon, and so on.
- Area is defined as the space within the perimeter of a flat object or figure.
- The measurement is done in square units, with square meters (m2) being the standard unit.
- The area is calculated using predefined formulae for squares, rectangles, circles, triangles, trapeziums, and so on.
- The area of regular polygons is simple to calculate since their dimensions are defined and known.
- For example, the area of a square can be simply calculated if we know the length of one of its sides, as all of its sides are equal.
- However, the area of an irregular polygon is determined as a combination of two or more regular polygons.
| Table of Content |
Key Terms: Polygon, Regular polygon, Irregular polygon, Rhombus, Triangle, Area of polygon, Parallelogram, Examples of polygon, Squares, Straight line, Curved line
What is Polygon?
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A polygon refers to a two-dimensional figure that has a closed formation, structured with a straight line and not curved lines.
- Common geometric figures like triangles, rectangles, squares, etc. are examples of polygons.
- Polygons are classified based on the regularity of their shape.
- The regularity of the shape of the plane figure means whether the length of the sides of the figures and the angles they form are symmetric.
Below are the different types of polygons.
Regular Polygons
Regular polygons are defined as having all of their sides and interior angles equal.
- Regular polygons include squares, rhombuses, and equilateral triangles.
- Regular polygons have not only congruent sides but also congruent angles. That indicates they are equiangular.
Irregular Polygons
An irregular polygon does not have equal sides or angles in measure. Irregular polygons include the scalene triangle, the right triangle, the isosceles triangle, the rectangle, the parallelogram, the irregular pentagon, and the irregular hexagon.
Polygons
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| Related Articles | ||
|---|---|---|
| Types of Polygon | Hexagon | Triangle |
| Pythagoras Theorem | Properties of Triangle | Quadrilateral and Types |
Area of Polygon
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The area of any given polygon whether it is a triangle, square, quadrilateral, rectangle, parallelogram, rhombus, hexagon, or pentagon, is defined as the space enclosed by it in a two-dimensional plane.
- The areas or formulas for the areas of different types of polygons vary depending on their shape.
- Assume that in order to calculate the area of a triangle, we must first determine the length of its base and height.
- Usually, the area is measured and represented in square units.
Calculation of Area of Polygon with n-sides
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The area of a regular polygon is calculated using different formulas constituted for each shape. For example, the area of the square is calculated by squaring the length of the side. Similarly, each regular polygon will have unique formulas to find its area.
- To calculate the area of a polygon that is not regular or whose formula is undetermined we divide it into triangles, squares, trapezoids, and so on.
- The objective is to view the given geometry as a collection of geometries for which we can calculate the area.
- We next calculate the area of each segment and combine them together to get the total area of the polygon.
The following are the steps to find the area of polygon with n-sides
- Observe the given irregular polygon as how many regular polygons are joined to form that figure.
- Draw lines and split the irregular polygon into two or more regular polygons.
- Find the area of the regular polygons thus formed, with the help of the formulas constituted to them.
- Find the sum of the areas of the regular polygons thus formed, which will be equal to the area of the irregular polygon.
Consider the example of a polygon shown below
Irregular polygon
This polygon can be divided into triangles and trapezium.
Method to find the area of an irregular polygon
We can calculate the area by using any of the above-mentioned divisions.
- In the first example, we may add the areas of the triangle and trapezium to get the area of the polygon.
- In the second, we combine the areas of three triangles to get the area of a polygon.
- In both cases, the result is the same.
Area of Polygon Formulas
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Here are some of the important formulas one should keep in mind when they are going to calculate the area of the polygon:
| Name of the Polygon | Area Formula |
|---|---|
| Square | (side)2 |
| Triangle | 1/2 x (base) x (height) |
| Rectangle | (Length) x (Breadth) |
| Pentagon | 5/2 x (side length) x (distance from the center of sides to the center of the pentagon) |
| Rhombus | 1/2 x (product of diagonals) |
| Hexagon | (3√3)/2 x (distance from the center of sides to the center of the hexagon) |
Solved Example
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Ques. Find the area of the polygon shown in the figure.
Ans. The above figure is an irregular polygon, so we first divide the given figure into regular polygons.
- AMIB is a rectangle
- BCGH is a square
- CDEM is a square
- JKL is a triangle
The total area of the figure is given by
A = Area (ABIM) + Area (BCGH) + Area (CDEF) + Area (JKL)
⇒ A = (AB × BI) + (BC × CG) + (CD × DE) + (1/2 × LJ × KO)
⇒ A = ( 10 × 5) + (3 × 3) + (2 × 2) + (1⁄2× 4 × 4)
⇒ A = 50 + 9 + 4 + 8
⇒ A = 71 cm2
Things to Remember
- Area is defined as the space within the perimeter of a flat object or figure.
- Regular polygons are defined as having all of their sides and interior angles equal.
- An irregular polygon does not have equal sides or angles in measure.
- The area of regular polygons is simple to calculate since their dimensions are defined and known.
- The area of an irregular polygon is determined as a combination of two or more regular polygons.
- To calculate the area of a polygon that is not regular or whose formula is undetermined we divide it into triangles, squares, trapezoids, and so on.
Also Read:
Sample Questions
Ques. What is a polygon? (1 Mark)
Ans. A polygon is a closed, flat object that consists of straight lines. Polygons include at least three sides and three angles. A polygon cannot have curved sides.
Ques. What are 3 examples of a polygon? (1 Mark)
Ans. Triangles, hexagons, pentagons, and quadrilaterals are all examples of polygons.
Ques. What is the difference between regular and irregular polygons? (3 Marks)
Ans. A regular polygon is a closed geometric figure which has equal sides and equal angles. The common regular polygon includes equilateral triangle, square, regular pentagon, etc. An irregular polygon is a closed geometric figure which doesn’t have equal angles or equal sides. This clearly states that the figure doesn’t have a standard form. But for the calculation of the area of an irregular polygon, the polygon will be divided into parts, making it into a collaboration of different regular polygons.
Therefore,
Area of irregular polygon = sum of areas of the regular polygon within the irregular polygon
Ques. A square PQRS has a side PQ= 3cm. Can you calculate the area of this polygon? (3 Marks)
Ans. We know that a square is a polygon that has equal sides.
Therefore, area of square = a2 (where ‘a’ is the length of any side)
Here, PQ = 3 cm
Therefore, the area of the square PQRS = 32= 9 cm2
Ques. A regular polygon’s area is 24 cm2. If the polygon has apothem 6cm, what would be its side’s length? Given that the polygon has 12 sides where the triangle can be divided into 12 congruent isosceles triangles. (3 Marks)
Ans. In the given situation, we have the following formula,
Area of the polygon = no. Sides of the polygon (½ x length of apothem x length of the side)
Here, 24 = 12 (1/2 x 6 x length of the side)
Solving the above we will get,
Length of the side = 0.67 cm
Ques. EFG is an equilateral triangle. If its perimeter is 18 cm, find the area of the polygon. Thus state that the parameter of a polygon is not equal to its area. (3 Marks))
Ans. We know that an equilateral triangle has all sides equal. Therefore,
The perimeter of triangle EFG = 3s
(where 's' stands for the length of each side)
This gives us that, s = 18/ 3 = 6 cm
We also know that area of the equilateral triangle = (√3/4) x s2
Here, area of triangle EFG = (√3/4) x 62
= 15.59 cm2
Ques. Rahul found that he had wrongly marked OC = 16 cm instead of OC = 19 cm. Keeping this fact in mind, can you help Rahul find the actual area of the polygon ABCD? The area of the figure as per Rahul was 384 cm2. (3 Marks)
Ans. The given figure is a Rhombus.
Area of Rhombus = 1/2 x (product of diagonals)
Here, OC = 19 cm (as rectified)
Therefore, the length of the diagonal AC = 19 x 2 cm
= 38 cm
Similarly, the length of BD = 24
Therefore, the Area of ABCD = 1/2 x 38 x 24
= 456 cm2
Ques. Find the area of the polygon ABCD, where AB = DC = 20 cm and AD = BC = 30 cm. The length of the shortest line drawn to AB from C is 15 cm. (3 Marks)
Ans. The given polygon ABCD is a parallelogram.
We know, the area of a parallelogram = length of its side x length of the shortest line drawn from the side.
Here, the area of the parallelogram ABCD = AB x CQ
= 15 x 20
= 300 cm2
Ques. The area of a Rhombus ABCD was given as 96 cm2. If AO = 8cm, find the length of OB. (here, AO= OC and BO = OD) (4 Marks)
Ans. Given that, ABCD is a kite, and AO = OC and BO = OD.
Therefore, the length of the diagonal AC = 16 cm
We know that,
Area of a Rhombus = 1/2 x product of its diagonals
Here, the area of ABCD = 1/2 x AC x BD
= 1/2 x 16 x BD
That is 96 cm2 = 1/2 x 16 x BD (given)
Solving the above, we will get
BD = 12 cm
Since BO = OD, BO = ½ x 12
= 6 cm
Ques. In a triangle, ABC, AC = 8cm, BC = 6 cm, and altitude BE = 4 cm. Find the length of another altitude AD. (hint: find the area of the triangle ABC, first) (4 Marks)
Ans. Here it is given that AC = 8cm, BC = 6 cm, and altitude BE = 4 cm.
We know that,
Area of a triangle = 1/2 x length of base x altitude from the base
Here, the area of triangle ABC = 1/2 x BC x BE
= 1/2 x 6 x 4
= 12 cm2
Considering the altitude AD and base AC we have
Area of triangle ABC = 1/2 x AD x AC
12 = ½ x AD x 8
Solving the above, we will get AD = 3 cm.
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