Areas Related to Circles: Area of a Circle, Semi Circle, Arc and Examples

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In mathematics, we have many geometrical figures. One of such is a circle. A circle can be defined as a two-dimensional figure that is closed. In a circle, a set of points in a plane is at the same distance from the centre. The distance that is between the centre and any other point in the circle is known as the radius. The distance from one edge of a circle to another, passing through the centre, is known as the diameter. The circumference of the circle is its perimeter.

Circle Calculator - Find Radius, Circumference, & Area - Inch Calculator

Table of Content

Area of a Circle
Length of an arc of a sector
Visualizations
Sample Questions

Key Terms: Circle, Area of Circle, Circumference of circle, Length of an arc, Segment of the circle, Sector of a circle

Area of a Circle

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The area of a circle can be calculated by πr²

Here, r is the radius of the circle and π=\(\frac{22}{7}\) i.e approximately 3.14

The video below explains this:

Areas Related to Circles Detailed Video Explanation:

Circumference of a Circle

The perimeter of a circle is the distance covered by the boundary of a circle. The perimeter of a circle has an exceptional name: Circumference, which is π times the width which is given by the formula 2πr.

Segment of a Circle

The segment is a region constrained by a chord of a circle and its arc. A segment with an intercepted arc less than a small circle is called a minor segment. A sector with an intercepted arc greater than a small circle is called a major segment.

Sector of a Circle

A sector of a circle is clarified as the region of a circle enclosed by an arc and two radii. The smaller the area is known as the minor sector while the larger area is regarded as the major sector.

The angle of a Sector

The angle of the sector is the angle that is surrounded by the two radii of the sector.

Length of an arc of a sector

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The length of the arc of a field can be calculated by using the expression for the circumference of a circle and the angle of the area, using the method formula:

L= \(\frac{θ}{360^o}\) × 2πr

Where θ is the angle of the area and r is the radius of the circle.

Visualizations

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Areas of different plane figures

Area of a square (side l) =l²

Area of a rectangle =l×b, where l and b are the length and breadth of the rectangle

The Area of a parallelogram = b×h, where “b” is the base, and “h” is the perpendicular height

How to find the area of a parallelogram - SSAT Middle Level Math

Area of a trapezium = \(\frac{1}{2}\)[(a+b)×h],

Where

h is the trapezium height

Areas of Combination of Plane figures

Let us find the area of the shaded portion in the following figure: Provided the ABCD is a square of side 30 cm and has 4 equivalent circles enclosed within it.

12.4 Areas of Combinations of Plane Figures | blind2visionary.org

Looking at the figure we can visualize that the shaded area

= A(square ABCD) − 4 ×A(Circle).

The Diameter for every circle is 15 cm.

=(l²)−4×(πr²)

=(28²)−[4×(π×49)]

=784−[4×22/7×49]

=784−616

=168cm²

Sample Questions

Ques: Define a circle? (1 mark)

Answer. A circle is a closed two-dimensional curve-shaped figure, where all the points on the circle are equidistant from the main point that is its centre.

Ques:What is the area of a circle whose perimeter is 52 cm? (1 mark)

Ans: Circumference of a circle = 2πr

From the question,

2πr = 52

Now, area of circle = πr² = π × (26/π)²

So, area of circle = (26×26)/π = 676 cm²

Ques: Calculate the sector of a circle whose perimeter is 31.4 cm. (2 marks)

Ans:

Given:

Perimeter = 31.4 cm

From the perimeter, the radius can be calculated:

2 π r = 31.4

(2)(3.14)r = 31.4

r = 31.4 /(2)(3.14)

r=10/2

A = π(5)2

A = 3.14 x 25

A = 78.5 cm²

Ques: State down some important circle formulas. (1 mark)

Ans: If “r” is the radius of the circle, the formula for the sector and the perimeter of a circle are:

The perimeter of a Circle = 2πr units

The area of a circle = πr² square units.

Ques: Define the radius and diameter of a circle. (1 mark)

Ans. The radius of a circle is the line segment that joins the center point and the circle’s external surface. The diameter is defined as the longest chord of any circle. The measurement of the diameter of a circle is twice its radius.

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CBSE X Related Questions

1.
Form the pair of linear equations for the following problems and find their solution by substitution method.
(i) The difference between two numbers is 26 and one number is three times the other. Find them.
(ii) The larger of two supplementary angles exceeds the smaller by 18 degrees. Find them.
(iii) The coach of a cricket team buys 7 bats and 6 balls for Rs 3800. Later, she buys 3 bats and 5 balls for Rs 1750. Find the cost of each bat and each ball.
(iv) The taxi charges in a city consist of a fixed charge together with the charge for the distance covered. For a distance of 10 km, the charge paid is Rs 105 and for a journey of 15 km, the charge paid is Rs 155. What are the fixed charges and the charge per km? How much does a person have to pay for travelling a distance of 25 km.
(v) A fraction becomes\(\frac{ 9}{11}\), if 2 is added to both the numerator and the denominator. If, 3 is added to both the numerator and the denominator it becomes \(\frac{5}{6}\). Find the fraction.
(vi) Five years hence, the age of Jacob will be three times that of his son. Five years ago, Jacob’s age was seven times that of his son. What are their present ages?

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2.
The following frequency distribution gives the monthly consumption of electricity of 68 consumers of a locality. Find the median, mean and mode of the data and compare them
Monthly consumption
(in units)
Number of consumers
65 - 85
4
85 - 105
5
105 - 125
13
125 - 145
20
145 - 165
14
165 - 185
8
185 - 205
4

Monthly consumption (in units)	Number of consumers
65 - 85	4
85 - 105	5
105 - 125	13
125 - 145	20
145 - 165	14
165 - 185	8
185 - 205	4

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3.
A vertical pole of length 6 m casts a shadow 4 m long on the ground and at the same time a tower casts a shadow 28 m long. Find the height of the tower.

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4.
Which of the following are APs? If they form an AP, find the common difference d and write three more terms.
(i) 2, 4, 8, 16, . . . .
(ii) \(2, \frac{5}{2},3,\frac{7}{2}\), . . . .
(iii) – 1.2, – 3.2, – 5.2, – 7.2, . . . .
(iv) – 10, – 6, – 2, 2, . . .
(v) 3, \(3 + \sqrt{2} , 3 + 3\sqrt{2} , 3 + 3 \sqrt{2}\) . . . .
(vi) 0.2, 0.22, 0.222, 0.2222, . . . .
(vii) 0, – 4, – 8, –12, . . . .
(viii) \(\frac{-1}{2}, \frac{-1}{2}, \frac{-1}{2}, \frac{-1}{2}\), . . . .
(ix) 1, 3, 9, 27, . . . .
(x) a, 2a, 3a, 4a, . . . .
(xi) a, \(a^2, a^3, a^4,\) . . . .
(xii) \(\sqrt{2}, \sqrt{8} , \sqrt{18} , \sqrt {32}\) . . . .
(xiii) \(\sqrt {3}, \sqrt {6}, \sqrt {9} , \sqrt {12}\) . . . . .
(xiv) \(1^2 , 3^2 , 5^2 , 7^2\), . . . .
(xv) \(1^2 , 5^2, 7^2, 7^3\), . . . .

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5.
Check whether \(6n\) can end with the digit \(0\) for any natural number \(n\).

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6.
Find the sums given below :
\(7 + 10\frac 12+ 14 + ....... + 84\)
\(34 + 32 + 30 + ....... + 10\)
\(–5 + (–8) + (–11) + ....... + (–230)\)

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Areas Related to Circles: Area of a Circle, Semi Circle, Arc and Examples

Area of a Circle