Perimeter and Area Class 7 Notes Maths Chapter 9

What is Perimeter and Area?

Area and perimeter are two fundamental concepts in geometry that describe the size and shape of a 2D object.

Perimeter

• The perimeter of a shape is the total length of its boundary.
• It is calculated by adding the lengths of all sides of the shape.
• Perimeter is measured in linear units such as meters (m), centimeters (cm), or inches (in).

Area

• The area of a shape is the space enclosed within its boundary.
• It is measured in square units such as square meters (m²), square centimeters (cm²), or square inches (in²).

Area of Parallelograms

A parallelogram is a two-dimensional geometrical shape whose sides are parallel to each other. It is a type of polygon having four sides (also called quadrilateral), where the pair of parallel sides are equal in length.

To find the area of a parallelogram, you can follow these steps:

1. Choose a base (any one side of the parallelogram).
2. Determine the height (the perpendicular distance from the base to the opposite side).
3. Use the formula: Area = Base × Height
   

Let us understand this with an example:

Example 1: Find the area of the following parallelograms: 

Ans: 
Base = 8 cm
Height = 3.5 cm
Area of parallelogram = Base × Height
Area of parallelogram = 8 cm x 3.5 cm = 28 cm²

Ans: 
Base = 8 cm
Height = 2.5 cm
Area of parallelogram = Base × Height
Area of parallelogram = 8 cm x 2.5 cm = 20 cm²

Example 2: PQRS is a parallelogram. QM is the height from Q to SR and QN is the height from Q to PS. If SR = 12 cm and QM = 7.6 cm. Find: 

(a) the area of the parallelogram PQRS 

(b) QN, if PS = 8 cm 

Ans: 
Given:  SR=12 cm, QM = 7.6 cm, PS = 8cm
(a) Area of parallelogram = base x height = 12 x 7.6 = 91.2 cm²
(b) Area of parallelogram = base x height
=> 91.2 = 8 x QN 
=> QN = 91.2/8 
= 11.4 cm.

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Area of Triangle

A triangle is a polygon with three vertices, and three sides or edges that are line segments. A triangle with vertices A, B, and C is denoted as ABC.

Triangle ABC

To find the area of a triangle within a parallelogram, you can follow these steps

1. Identify the Triangle

• Look at one of the triangles inside the parallelogram.
• The base of the triangle is one side of the parallelogram.
• The height is the straight distance from the base to the opposite side.

2. Understand the Relationship

• A parallelogram is made up of two identical triangles.
• Therefore, the area of the parallelogram is twice the area of one triangle.

3. Calculate the Area of the Parallelogram

• Use the formula: Area = Base × Height
  

4. Apply the Relationship for the Triangle

• Since the parallelogram has two identical triangles, the area of one triangle is half the area of the parallelogram.

12×Are

Parallelogram ABCD

Area of each triangle = $\frac{1}{2}\backslash frac\{1\}\{2\}$ (Area of parallelogram)

$=\frac{1}{2}=\; \backslash frac\{1\}\{2\}$= (base × height) (Since area of a parallelogram = base × height)

$=\frac{1}{2}(b\times h)=\; \backslash frac\{1\}\{2\}\; (b\; \backslash times\; h)$=(or $\frac{1}{2}bh\backslash frac\{1\}\{2\}\; bh$, in short)

All the congruent triangles are equal in area but the triangles equal in area need not be congruent.

Example 1: Find BC, if the area of the triangle ABC is 36 cm² and the height AD is 3 cm.

Ans: 
Height = 3 cm, Area = 36 cm²
Area of the triangle ABC = 1/2 x b x h
=> 36 = 1/2 x b x 3  => b = 24 cm
Base BC = 24 cm

Ans:  
In triangle ABC, AD = 6cm and BC = 9cm
Area of triangle = 1/2 x base x height = 1/2 x AB x CE
=>  27 = 1/2 x 7.5 x CE  
=>   CE = (27 x 2) /7.5  => CE = 7.2 cm
Height from C to AB ie.., CE is 7.2 cm

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Circles

A circle is defined as a collection of points on a plane that are at an equal distance.

Circle

• Diameter: Any straight line segment that passes through the centre of a circle and whose end points are on the circle is called its diameter.
• Radius: Any line segment from the centre of the circle to its circumference.

Circumference of Circle

The distance around a circular region is known as its circumference.

where,

• r is the radius of the circle
• π is an irrational number, whose value is approximately equal to 3.14 or it can be taken as 22/7

Diameter(d) of a circle is equal to twice the radius(r) =2r 
Therefore, Circumference = Diameter x π 
C = π d

Note: Circles with the same centre but different radii are called concentric circles. 

Example: If the radius of the circle is 25 units, find the circumference of the circle. (Take π = 3.14)

Ans: 
Given, radius = 25 units
Let us write the circumference formula and then we will substitute the value of r (radius) in it.
Circumference of circle formula = 2πr
C = 2 × π × 25
C = 2 × 3.14 × 25 = 157 units
Therefore, the circumference of a circle is 157 units.

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Area of Circle

The area of a circle is the region enclosed in the circle.

where 

• r is the radius of the circle
• A is the area of the circle 
• π is an irrational number, whose value is approximately equal to 3.14 or it can be taken as 22/7

$A\; =\; \backslash frac\{C^2\}\{4\backslash pi\}$

Example 1: The radius of a circular pipe is 10 cm. What length of a tape is required to wrap once around the pipe (π = 3.14)?

Ans: 
Radius of the pipe (r) = 10 cm
 Length of tape required is equal to the circumference of the pipe. Circumference of the pipe = 2πr = 2 × 3.14 × 10 cm = 62.8 cm
Therefore, length of the tape needed to wrap once around the pipe is 62.8 cm.

Example 2: Find the perimeter of the given shape  (Take π = 22/7 ).

Ans: In this shape, we need to find the circumference of semicircles on each side of the square.Circumference of the circle = πd
Circumference of the semicircle = 1/2 πd = 1/2 x 22/7 × 14 cm = 22 cm Circumference of each of the semicircles is 22 cm .

Therefore, the perimeter of the given figure = 4 × 22 cm = 88 cm.

Example 3: Diameter of a circular garden is 9.8 m. Find its area. 

Ans: 
Diameter, d = 9.8 m
Therefore, radius r = 9.8 ÷ 2 = 4.9 m
Area of the circle = πr² = 22/7 x (4.9)² m² = 22/7 × 4.9 x 4.9 m²= 75.46 m²

Example 4:The adjoining figure shows two circles with the same centre. The radius of the larger circle is 10 cm and the radius of the smaller circle is 4 cm. Find:
 
(a) the area of the larger circle 
(b) the area of the smaller circle 
(c) the shaded area between the two circles. (π = 3.14)

Ans: 
(a) Radius of the larger circle = 10 cm So, area of the larger circle = πr² = 3.14 × 10 × 10 = 314 cm²
(b) Radius of the smaller circle = 4 cm Area of the smaller circle = πr² = 3.14 × 4 × 4 = 50.24 cm²
(c) Area of the shaded region = (314 – 50.24) cm² = 263.76 cm²