Perimeter of the Rectangle and Square: The perimeter is defined as the length of the boundary of a closed figure. We usually use terms to find the perimeters are length and breadth. The various measurement units of length, like kilometres, metres, centimetres, millimetres, decimetres, inches, feet, miles, etc., can be used to denote perimeter.

Perimeter = 2(length + width) is twice the sum of its length and breadth and is computed using the formula: Perimeter = 2(length + width). This article will learn about the area and perimeter of rectangles and squares, along with some solved examples. We have also provided the CBSE Sample papers for your reference. 

What is Perimeter?

Perimeter is the distance covered along the boundary of a closed figure when you go round the figure once.  The shape’s perimeter 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 shape round.

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

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.

Rectangle & Square: Definition, Shape & Properties

The definition, shape and properties of Rectangle and Square are discussed below:

Rectangle

The rectangle is one of the quadrilaterals in which all four angles are right angles, and opposite sides are parallel and equal to each other.

Properties of a Rectangle

1. Rectangles have four sides and four vertices.
2. Rectangles have four right angles.
3. The opposite sides of a rectangle are congruent.
4. The opposite sides of a rectangle are parallel to each other.
5. Diagonals of a rectangle bisect each other.

Square

A regular quadrilateral, in which all four sides are of equal lengths and all four equal angles measuring \({90^ \circ }\) is known as a square.

Properties of a Square

1. All four sides of a square are congruent
2. All four angles of a square are right angles
3. Diagonals of a square bisect each other, and they are perpendicular
4. Consecutive angles of a square are supplementary

Perimeter of Square and Rectangle?

The sum of all the sides of a rectangle is known as its perimeter and the sum of all sides of a square is known as its perimeter.

Formula of Perimeter of Rectangle

Consider a rectangle as shown below. Let \(l\) and \(b\) denote its length and breadth, respectively. If \(P\) denotes perimeter, then

\(P = AB + BC + CD + DA\)
\( \Rightarrow P = l + b + l + b\)
\(\Rightarrow P = 2l + 2b\)
\( \Rightarrow P = 2(l + b)\)
\( \Rightarrow P = 2({\rm{length + breadth}})\)

Formula of Perimeter of Square

We know that a square is a rectangle whose length and breadth are equal. Therefore, the formula obtained for the perimeter of a rectangle may be used to obtain the perimeter of a square. Since all four sides are equal, we may use the following simpler forms. Let \(a\)  be the length of each side of a square and if \(p\)  denotes the perimeter, then.

\(P = AB + BC + CD + DA\)
\( \Rightarrow P = a + a + a + a\)
\( \Rightarrow P = 4a\)
So, the perimeter of a square is \(4\) times the length of its side.

Solved Questions

Q.1. The length and breadth of a park are in the ratio \(2:1\), and its perimeter is \(240\,m.\) A path of \(2\,m\) wide runs inside it, along the boundary. Find the cost of paving the path at \(₹3\,{\rm{per}}\,{{\rm{m}}^{\rm{2}}}\)
Ans: Let \(ABCD\) be the park, whose length \(AB\) and breadth \(AD\) are in the ratio \(2:1\). Let the shaded region represent the \(2\,m\) inside path that runs inside the park \(ABCD\). Since length and breadth of the park are in the ratio \(2:1\), let the length and breadth of the park be \(2x\) and \(x\) respectively. Then,

Perimeter \( = 2(2x + x)\)
But, perimeter \( = 240\;{\rm{m}}\)
Therefore, \(2(2x + x) = 240{\rm{m}}\)
\( \Rightarrow 6x = 240\;{\rm{m}}\)
\( \Rightarrow x = \frac{{240}}{6} = 40\;{\rm{m}}\)
Therefore, length \(AB = 2 \times 40 = 80\;{\rm{m}}\) and breadth \(AD = 4{\rm{0m}}\)
We have \(PQ = (80 – 2 – 2){\rm{m = 76m}}\) and \(PR = (40 – 2 – 2){\rm{m}} = 36\;{\rm{m}}\)
So, area of path = Area of rectangle \(ABCD\) – Area of rectangle \(PQRS\)
\( = (80 \times 40 – 76 \times 36){{\rm{m}}^2}\)
\( = (3200 – 2736){{\rm{m}}^2}\)
\( = 464\;{{\rm{m}}^2}\)
Hence, cost of paving the path \( = ₹(464 \times 3) = ₹1392\)
Therefore, the cost of paving the path is \(₹1392\).

Q.2.  A wire is in the shape of a square of side \( = 10\,{\rm{cm}}{\rm{.}}\) If the wire is rebent into a rectangle of length \(12\,{\rm{cm}},\) find its breadth. 
Ans: Given side of the square\( = 10\,{\rm{cm}}\)
Length of the wire = Perimeter of the square \( = 4 \times {\rm{side}} = 4 \times 10\;{\rm{cm}} = 40\;{\rm{cm}}\)
\(l=\) Length of the rectangle \({\rm{12cm}}\)
Let \(b\) be the breadth of the rectangle.
Now, 
Perimeter of the rectangle = Perimeter of the square
\( \Rightarrow 2(l + b) = 40\)
\( \Rightarrow 2(12 + b) = 40\)
\( \Rightarrow 24 + 2b = 40\)
\( \Rightarrow 2b = 40 – 24\)
\( \Rightarrow 2b = 16\)
\( \Rightarrow b = 8\;{\rm{cm}}\)
Therefore, breadth of the rectangle \( = 8\,{\rm{cm}}\)

Q.3. Find the perimeter of a rectangle whose length and breadth are \({\rm{25}}\,{\rm{m}}\) and \({\rm{15}}\,{\rm{m}}\) respectively.
Ans: Let \(l\) be the length of the rectangle and \(b\) be the breadth of the rectangle.
\(l = 25\,{\rm{m}}\) and \(b = 15\,{\rm{m}}\)
We know that, the perimeter of a rectangle \( = 2(l + b)\)
\( = 2(25 + 15){\rm{m}}\)
\( = (2 \times 40){\rm{m}}\)
\( = 80\;{\rm{m}}\)
Therefore, the perimeter of a rectangle is \( = 80\;{\rm{m}}\)

Q.4. Find the breadth of a rectangle whose perimeter is \({\rm{360}}\,{\rm{cm}}\) and whose length is \({\rm{100}}\,{\rm{cm}}{\rm{.}}\)
Ans: Let \(l\) be the length of the rectangle, \(b\) be the breadth of the rectangle and \(P\) is the perimeter of a rectangle.
Given: \(P = 360\;{\rm{cm}},l = 100\;{\rm{cm}}\) and \(b=?\)
We know that the perimeter of a rectangle \(P = 2(l + b)\)
\( \Rightarrow 360 = 2(100 + b){\rm{cm}}\)
\( \Rightarrow \frac{{360}}{2} = 100 + b\)
\( \Rightarrow 180 – 100 = b\)
\( \Rightarrow b = 80\;{\rm{cm}}\)
Therefore, the breadth of the rectangle is \(80\;{\rm{cm}}\)

Q.5. Find the perimeter of a square whose side is \(5\,{\rm{cm}}\)
Ans: Let \(a\) be the length of the side of the square.
We know that the perimeter of a square \( = 4a\)
\( \Rightarrow {\rm{Perimeter}} = 4 \times 5\;{\rm{cm}}\)
\( \Rightarrow {\rm{Perimeter}} = 20\;{\rm{cm}}\)
Therefore, the perimeter of the square is \(20\,{\rm{cm}}\)

Q.6. Pinky runs around a square field of side \(75\,{\rm{m,}}\) Bobby runs around a rectangular field with a length of \(160\,{\rm{m}}\) and breadth of \(105\,{\rm{m}}{\rm{.}}\) Who covers more distance by how much?
Ans: Distance covered by Pinky in one round \(=\) Perimeter of the square field of side \(75\,{\rm{m}}\)
\( \Rightarrow 4 \times \)length of a side
\( = 4 \times 75\;{\rm{m}} = 300\;{\rm{m}}\)
Now, distance covered by Bobby in one round \(=\) Perimeter of a rectangular field
\( = 2 \times ({\rm{length + breadth}})\)
\( = 2 \times (160\;{\rm{m}} + 105\;{\rm{m}})\)
\( = 2(265\;{\rm{m}}) = 530\;{\rm{m}}\)
Therefore, the difference in distance covered \( = 530\;{\rm{m}} – 300\;{\rm{m}} = 230\;{\rm{m}}\)
Hence, Bobby covers more distance and by \(230\;{\rm{m}}\)

Summary

This article discussed the definition and examples of perimeter, definition, and properties of rectangle and square. We have understood the definition of perimeter of rectangle and square and formulas to find the perimeter of square and rectangle and solved some problems regarding the same.

FAQs

Q.1. What is a perimeter in maths?
Ans: 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.

Q.2.What is the area and perimeter of square and rectangle?
Ans: If a be the length of each side of a square and if \(P\) denotes the perimeter, then
\(P = 4a.\) .
If \(l\) and \(b\) denote its length and breadth, respectively. If \(P\) represents perimeter, then \(P = 2(l + b)\)
If \(l\) and \(b\) denote its length and breadth, respectively. The area of rectangle \(A\) is given by \(A = (l \times b)\)
If a be the length of each side of a square and if \(A\) denotes the perimeter, then area of a square \(A = {a^2}\)

Q.3. What is the formula of perimeter of triangle?
Ans: The perimeter of a triangle with the length of their sides \(a, b\) and \(c\) units is given by \(P = a + b + c\) units.

Q.4. Explain perimeter of rectangle and square?
Ans: A rectangle is a quadrilateral in which all four angles are right angles, and opposite sides are parallel and equal to each other. A square is a rectangle with all four sides of equal lengths and all four equal angles measuring \({90^ \circ }\). The sum of all the sides of a rectangle is known as its perimeter. The sum of all sides of a square is known as its perimeter.

Q.5. What is a formula of perimeter of rectangle?
Ans: If \(l\) and \(b\) denotes the length and breadth, respectively of a rectangle and if \(P\) represents the perimeter, then the formula of perimeter of the rectangle is given by
\(P = 2(l + b)\)

Check Properties of Rectangle Here

a. Class 8 Maths Practice Questions

b. Class 9 Maths Practice Questions

c. Class 10 Maths Practice Questions

d. Class 11-12 Maths Practice Questions

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