Area Enclosed By Circle

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Introduction

Circle is locus of a point which moves at a constant distance from a fixed point. This constant distance is called radius of a circle and the fixed point is called centre of the circle. The general equation of a circle is,

x2 + y2 + 2gx + 2fy + c = 0

where, g, f and c are constants.

For this circle, centre is at (-g,-f) and radius is calculated as \[\sqrt{g^{2}+f^{2}-c}\]. If the centre of the circle is at origin, then the equation of the circle is given by,

x2 +y2 = r2

where, r is radius.          

Diameter of a circle is a line passing through the centre of a circle touching two points on the circle. Diameter of a circle is double its radius. 


Formula for Area of a Circle

Consider the following figure,

[Image will be uploaded soon]

The area of this circle is given as a product of pi and the square of radius. Therefore,

Area = πr2

The value of π can be taken as 22/7 or 3.14.


Derivation

Cut a circle into equal sectors as follows:

[Image will be uploaded soon]

Arrange these sectors in a straight line as follows:

[Image will be uploaded soon]

This forms a rectangle. The area of this rectangle will be equal to the area of circle. Area of a Rectangle is given by,

Area = length × breath 

Length of the rectangle is radius of the circle. The breadth of the rectangle is half the circumference of the circle. Circumference of a circle is given by, , where, r is radius of the circle. Therefore, Breadth of the Rectangle is:

\[\frac{2πr}{2}\] = πr

Area of the Rectangle is calculated as:

Area = length × breath 

Area = r × πr

Area = πr2

Therefore, Area of a Circle is given by,

πr2


Area of a Circle using Integrals in Calculus

Consider the following Circle,

x2 +y2 = a2

Here, radius = a

Hence, a diagram can be illustrated as follows:

[Image will be uploaded soon]

OB = OA = a

Since, the circle is divided into four equal sectors, area of the circle will be 4 times the area of OAB.

Therefore,

Area of Circle = \[4×\int_{0}^{a}ydx\]

Now,

x2 +y2 = a2

y = ±\[\sqrt{a^{2}-x^{2}}\]

Since, OAB lies in the first quadrant, y will be positive.

Therefore,

y = \[\sqrt{a^{2}-x^{2}}\]

Substituting this value in area,

Area of Circle = \[4×\int_{0}^{a}\sqrt{a^{2}-x^{2}}dx\]

Using, \[\sqrt{a^{2}-x^{2}}dx\] = \[\frac{1}{2}\sqrt{a^{2}-x^{2}}\] + \[\frac{a^{2}}{2}sin^{-1}\frac{x}{a}\] + c

Area is calculated as:

= \[4 [\frac{x}{2}\sqrt{a^{2}-x^{2}} + \frac{a^{2}}{2}sin^{-1}\frac{x}{a}]_{0}^{a}\]

= \[4 [(\frac{a}{2}\sqrt{a^{2}-a^{2}} + \frac{a^{2}}{2}sin^{-1}\frac{a}{a})-(\frac{0}{2}\sqrt{a^{2}-0^{2}} + \frac{0^{2}}{2}sin^{-1}\frac{0}{a})]\]

= \[4 [0 + \frac{a^{2}}{2}sin^{-1}(1)-0-0]\]

= \[4 × \frac{a^{2}}{2}sin^{-1}(1)\]

= \[4 × \frac{a^{2}}{2}× \frac{π}{2}\]

= πa2

Solved Examples

Example 1:

If Radius of a circle is 7 cm. What is its Area?

Solution:

Given,

Radius = 5 cm

Area of a circle is calculated as:

Area = πr2

Area = \[\frac{22}{7}\] × 7 × 7

Area = 154cm2

Therefore, area of the circle is 154cm2

Example 2:

If Diameter of a circle measures 70 cm. What is the Area of the Circle?

Solution:

Given,

Diameter = 70 cm

Therefore, radius = \[\frac{70}{2}\] = 35 cm 

Area of a circle is calculated as:

Area = πr2

Area = \[\frac{22}{7}\] × 35 × 35

Area = 3850cm2

Therefore, area of the circle is 3850cm2.

Example 3:

If Perimeter of the Circle is 80 cm, find its Area.

Solution:

Given,

Perimeter = 80 cm.

Therefore,

2πr = 80

r = \[\frac{80×7}{2×22}\]

r = 12.7 cm

Radius of the circle is 12.7 cm.

Area of a circle is calculated as:

Area = πr2

Area = 3.14 × (12.7)2

Area = 506.45 cm2

Therefore, area of the circle is 506.45 cm2.

Example 4:

A circular field has a circumference of 300 m. What is the cost of installing a flower bed at the rate of Rs 3.00 per m2?

Solution:

Given,

Circumference = 300 m

Therefore,

2πr = 300

r = \[\frac{300}{2×3.14}\]

r = 47.7 m

Radius of the circular field is 47.7 m.

Area of a circular field is calculated as:

Area = πr2

Area = 3.14 × (47.7)2

Area = 7144.4 m2

Therefore, area of the circular field is 7144.4 m2.

Cost of installing a flower bed is Rs 3.00 per m2.

Cost of installing a flower bed for 7144.4 m2 = 7144.4  × 3 = 21433.23

Therefore, cost of installing a flower bed in the circular field is Rs 21433.23.

Example 5:

Two tangents, OA and OB are drawn from the origin to the circle x2 + y2 + 2gx + 2fy + c = 0 and C is the centre of the circle. What is the Area of Quadrilateral OACB?

Solution:

A diagram can be illustrated as follows:

[Image will be uploaded soon]

Area of quadrilateral OACB = 2 × 1/2 × OA × AC

OA is length of tangent. Length of a tangent is given by \[\sqrt{S}\],where        

S = x2 + y2 + 2gx1 + 2fy1 + c .

Since, the tangent is drawn from origin, (x,y) = (0,0)

Therefore,

S = c.

Length of tangent = OA = \[\sqrt{S}\] = \[\sqrt{c}\]

AC is radius. Hence, it is given by,

\[\sqrt{g^{2}+f^{2}-c}\]

Therefore,

Area of quadrilateral OACB is,

2 × 1/2 × OA × AC

= \[\sqrt{c}.\sqrt{g^{2}+f^{2}-c}\]


Did You Know

  • The general equation of a second degree ax2 + by2 + 2hxy + 2gx + 2fy + c = 0represents a circle if a = b ≠ 0and h = 0.

  • The ratio of circumference and diameter of a circle is always constant, that is 22/7 or π.

  • The area between two radii of a circle is called sector. Area of a sector is given by, 

\[\frac{θ}{360°}\]× πr2

where, θ is the angle between the two radii of the circle.

  • A sector without the triangle formed by the given radii is called a segment. The area of a segment is calculated by subtracting area of triangle formed by the two radii from the area of the sector formed by the two radii.

  • The diameter of a circle circumscribed on a rectangle is equal to the diagonal of the rectangle.

  • When a square is circumscribed on a circle, the side of the square is equal to the diameter of the circle.

  • If a circle does not enclose any area, its radius is 0, and the circle is called a point circle.

FAQ (Frequently Asked Questions)

1. How is the Area of a Circumcircle of a Triangle Obtained?

Ans. A circle circumscribed on a triangle is called the circumcircle of the triangle. If the triangle is a right angled triangle, that is, if it satisfies the Pythagoras theorem, the hypotenuse of the triangle will form the diameter of the circle. Length of hypotenuse can be obtained by distance formula. The radius can be calculated by dividing the diameter by 2. Area of the circle can be calculated by multiplying pi and r2.

2. What is the difference between Circumference and Area of a Circle?

Ans. The length of the curved surface is called the circumference of the circle. Circumference is also called the perimeter. The space enclosed by a circle is called the area. Circumference is given by, 2πr whereas area is given by πr2 , where r is the radius. Moreover, the unit of area is squared, for example m2 or cm2 unlike units of perimeter.