Perimeter and Area: Definition, Important Formulas, Examples

Area and perimeter are the two important characteristics of two-dimensional shapes in Mathematics. Children of lower classes often get confused between Perimeter and Area. The area is defined as the amount of space occupied by any shape in two dimensions. On the other hand, Perimeter is the boundary or outline of a flat shape. This definition is applicable to any 2d shape of any size, whether it is regular or irregular. Every shape has its own unique formula for its area and perimeter. In this article, we discuss the differences between these two terms, the area and perimeter formulas for different shapes such as triangle, square, rectangle, circle, sphere, etc. Read on to learn more.

Perimeter and Area Definition

Perimeter: Perimeter is the distance covered along the boundary of a closed figure when you go round the figure once. The perimeter of a shape is defined as the overall distance around the shape. It is the length of any shape that can be expanded in a linear form. The perimeter of different figures can be equal in measure depending upon the measurements. For example, imagine a rectangle made of a wire of length \(L,\) the same wire can be reused to make a square, considering that all the sides are equal in length.

In the figure below, if we start from point \(P\) and move along the line segment, we again reach point \(P.\) We have made a complete round of the shape.

The distance covered is equal to the length of the wire used to draw the figure. Thus, the distance is known as the perimeter of the closed figure.

Area: The amount of surface enclosed by a closed figure is called its area. The area of a shape is usually determined with the help of its length and breadth. The area of a shape is a two-dimensional quantity. Hence, it is calculated in square units like square inches or square feet, square yard, etc.

Most of the shapes have edges and corners. The length and breadth of these edges are taken while calculating the area of a specific shape.

Conversion of Units

A unit can be converted into the other by using the following conversions:

(i) \(1~{\text{kilometre}}\left({{\text{km}}} \right) = 1000~\,{\text{metre}}\left({\text{m}} \right)\)
(ii) \(1\,{\text{metre}}\left({\text{m}} \right) = 100\,{\text{centimeters}}\left( {{\text{cm}}} \right)\)
(iii) \(1\,{\text{decimetre}}\left({{\text{dm}}} \right) = 10\,{\text{centimeters}}\left( {{\text{cm}}} \right)\)
(iv) \(1\,{\text{centimetres}}\left({{\text{cm}}} \right) = 10\,{\text{millimeters}}\left( {{\text{mm}}} \right)\)
(v) \(1{\mkern 1mu} {\rm{foot}}\,{\rm{ = }}\,{\rm{12}}{\mkern 1mu} {\rm{inches}}\)
(vi) \(1{\mkern 1mu} {\rm{yard}}\,{\rm{ = }}\,{\rm{3}}{\mkern 1mu} {\rm{feet}}\)
(vii) \(22{\mkern 1mu} {\rm{yard}}\,{\rm{ = }}\,{\rm{1}}{\mkern 1mu} {\rm{chain}}\)

Formula for Perimeter of Plane Figures

Perimeter of a square: A square is a plane figure with four equal straight sides and four right angles.

So, the perimeter of a square with the length of each side \(a\) units is given by \(4a\,{\text{units}}.\)

Perimeter of a rectangle: A plane figure with parallel sides equal to each other and four right angles is called a rectangle.

So, the perimeter of a rectangle with length \(l\,{\text{units}}\) and breadth \(b\,{\text{units}}\) units is given by \(2\left( {l + b} \right)\,{\text{units.}}\)

Perimeter of a parallelogram: A parallelogram is a quadrilateral whose each pair of opposite sides are parallel.

So, the perimeter of a parallelogram with length \(a\,{\text{units}}\) and width \(b\,{\text{units}}\) is given by \(2\left({a + b} \right)\,{\text{units}}{\text{.}}\)

Perimeter of a rhombus: A special case of a parallelogram, with four equal sides and opposite sides parallel with opposite angles are equal, is a rhombus.

So, the perimeter of a rhombus with side \(a\,{\text{units}}\) is given by \(4 a\,{\text{units}}{\text{.}}\)

Perimeter of a circle: A closed two-dimensional figure in which the set of all the points in the plane is equidistant from a centre is called a circle. 

So, the perimeter of a circle is nothing but its circumference. So, the perimeter of a circle with radius \(r\) is given by \(2\pi r\,{\text{units}}{\text{.}}\)

Perimeter of a triangle: A triangle is a three-sided figure with three edges and three vertices in which the sum of the interior angles is equal to \({180^ \circ }.\)

So, the perimeter of a triangle with the length of their sides \(a,b\) and \(c\,{\text{units}}\) is given by \(P = a + b + c\,{\text{units}}{\text{.}}\)

Perimeter of an equilateral triangle: An equilateral triangle is a triangle whose all three sides have equal lengths and have an equal angle.

So, the perimeter of an equilateral triangle with the length of the sides \(a\) units is given by \(P = 3a\,{\text{units}}.\)

Formula for Area of Plane Figures

(i) Area of a square: The area of a square with the length of each side \(a\,{\text{units}} = {a^2}\,{\text{sq}}{\text{.units}}{\text{.}}\)

(ii) Area of rectangle: The area of a rectangle with length \(l\) and breadth \(b\) is given by \(l \times b\,{\text{sq}}{\text{.units}}\)

(iii) Area of a parallelogram: The area of the parallelogram with base \(b\) and height \(h\) is given by \(b \times h\,{\text{sq}}{\text{.units}}\)

(iv)Area of a rhombus: Area of rhombus \( = \frac{1}{2} \times \left( {{\rm{product}}\,{\rm{of}}\,{\rm{length}}\,{\rm{of}}\,{\rm{diagonals}}} \right)\,{\rm{sq}}{\rm{.}}\,{\rm{units}}\)

(v) Area of a circle: The area of a circle with radius \(r\,{\text{units}}{\text{.}}\) is given by \(\pi {r^2}\,{\text{sq}}{\text{.units}}\)

(vi) Area of a triangle: The area of a triangle is given by \(\frac{1}{2} \times {\rm{base}} \times {\rm{height}}\,{\rm{sq}}{\rm{.}}\,{\rm{units}}\)

(vii) Area of an equilateral triangle: The area of an equilateral triangle with side \({\text{a}}\,{\text{units}}\) is given by \(\frac{{\sqrt 3 }}{4}{a^2}\,{\text{sq}}{\text{.units}}{\text{.}}\)

Table for Perimeter and Area Formulas

Diagram	Shape	Area	Perimeter
	Rectangle	\(a \times b\)	\(2\left({a + b} \right)\)
	Square	\({a^2}\)	\(4a\)
	Triangle	\(\frac{1}{2} \times b \times h\)	\(a + b + c\)
	Parallelogram	\(b \times h\)	\(2\left({a + b} \right)\)
	Circle	\(\pi {b^2}\)	\(2\pi b\)

Solved Examples – Perimeter and Area

Q.1. An athlete takes 10 rounds of a rectangular park, \(50\,{\text{m}}\) long and \(25\,{\text{m}}\) wide. Find the total distance covered by him.

Ans: Given: Length of the rectangular park \( = 50\,{\text{m}}\)
Breadth of the rectangular park \( = 25\,{\text{m}}\)
The total distance covered by the athlete in one round will be the perimeter of the park.
Now, the perimeter of rectangular park \( = 2\left({l + b} \right)\)
\( = 2\left({50 + 25} \right){\text{m}}\)
\( = 2\left({75} \right){\text{m}} = 150\,{\text{m}}\)
So, the distance covered by the athlete in one round is \(150\,{\text{m}}.\)
Therefore, distance covered by the athlete in \(10\) rounds \(10\, \times 150{\text{m=1500}}\,{\text{m}}\)
Therefore, the total distance covered by the athlete is \({\text{1500}}\,{\text{m}}{\text{.}}\)

Q.2. Find the cost of fencing a rectangular park of length \( 250\,{\text{m}}\) and breadth \( 175\,{\text{m}}\) at the rate of \(₹12\,{\rm{per}}\,{\rm{metre}}.\)

Ans: Length of the rectangular park \(=250\,{\rm{m}}\)
The breadth of the rectangular park \( = 175\,{\rm{m}}\)
To calculate the cost of fencing, we require a perimeter.
We know that the perimeter of rectangle \( = 2\left({l + b} \right)\)
So, area of rectangular park \( = 2\left({250 + 175} \right){\text{m}} = 2\left({425} \right){\text{m}} = 850\,{\text{m}}\)
Cost of fencing \(1\,{\text{m}}\) of park \( = ₹ 12\)
Therefore, the total cost of fencing the park \(= ₹ 12 \times 850 =₹ 10200\)

Q.3. A door frame of dimensions \(3\,{\text{m}} \times 2\,{\text{m}}\) is fixed on the wall of dimension \(10\,{\text{m}} \times 10\,{\text{m}}{\text{.}}\) Find the total labour charge for painting the wall if the labour charges for painting \(1\,{{\text{m}}^2}\) of the wall is \(₹ 2.50.\)

Ans: Painting of the wall has to be done, excluding the area of the door. Area of the door \( = l \times b\)
\( = 3 \times 2 = 6\,{{\text{m}}^2}\)
Area of wall including door \( = 10\,{\text{m}} \times 10\,{\text{m}} = 100\,{{\text{m}}^2}\)
Area of wall excluding door \( = \left({100 – 6} \right)\,{{\text{m}}^2} = 94\,{{\text{m}}^2}\)
Total labour charges for painting the wall\(= ₹ 2.50 \times 94 = ₹ 235\)

Q.4. Find the base of a triangle, if the area of a triangle is \(36\,{\text{c}}{{\text{m}}^2}\) and height is \(3\,{\text{cm.}}\)

Ans: Given: Area of a triangle \(36\,{\text{c}}{{\text{m}}^2,}\) height \(h = 3\,{\text{cm}}{\text{.}}\)
We know that, area of a triangle \( = \frac{1}{2} \times b \times h\)
\( \Rightarrow 36 = \frac{1}{2} \times b \times 3\)
\( \Rightarrow b = \frac{{72}}{3}\,{\text{cm}}\)
\( \Rightarrow b = 24\,{\text{cm}}\) Therefore, the base of a triangle is \(24\,{\text{cm}}{\text{.}}\)

Q.5. The perimeter of a regular hexagon is \(18\,{\text{cm}}{\text{.}}\) How long is its one side?

Ans: Perimeter of a regular hexagon \( = 18\,{\text{cm}}{\text{.}}\)
A regular hexagon has \(6\) sides, so we can divide the perimeter by \(6\) to get the length of one side.
One side of hexagon \( = 18\,{\text{cm}} \div 6 = 3\,{\text{cm}}.\)
Therefore, the length of each side of the regular hexagon is \(3\,{\text{cm}}.\)

Summary

In this article, we have learned about the definition of area and perimeter. Also, we have learned the different formulas to find the area and perimeter of different shapes and solved some practice problems.

Frequently Asked Questions (FAQ) – Perimeter and Area

The answers to some of the commonly asked questions about Perimeter and area are provided here:

Q.1. How do you find the perimeter and the area?
Ans: We have different formulas for different shapes to find the perimeter and the area.
Example: Perimeter of square with the length of the side a=4aunits
Area of the square with the length of the side a=a2sq.units

Q.2. What is the formula for the area and perimeter of a circle?
Ans: The area of the circle with radius runits is πr2 and perimeter of the circle is 2πr.

Q.3. What is the example for a perimeter?
Ans: The idea of the perimeter is widely used in our daily life.

1. A farmer who wants to fence his field.
2. A person preparing a track to conduct sports.
   All the above examples use the idea of the perimeter.

Q.4. What is the formula for the area and perimeter of a triangle?
Ans: Area of triangle =12×b×h where, b=base and h=height
The perimeter of a triangle with the length of their sides a,b and c units is given by p=a+b+cunits.

Q.5. What is the perimeter formula?
Ans: Perimeter is the distance covered along the boundary forming a closed figure when you go round the figure once. The perimeter of a shape is defined as the overall distance around the shape. Thus, it is the length of any shape that can be expanded in a linear form.

So, the formula of the perimeter depends on the shape and measurement of the given shape or object.
Example: Perimeter of a square with the length of each side a units is given by 4aunits.
The perimeter of a rectangle with length lunits and breadth bunits is given by 2(l+b)units.
The perimeter of a parallelogram with length aunits and width bunits is given by 2(a+b)units.

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