Question

Derive the formula of Area of sector and Area of segment .

Class 10th
Chapter - Area related to circles.

No Spam​

Answer 1

Answer:

☯︎Area of a Segment of a Circle Formula:-

➪The formula to find segment area can be either in terms of radians or in terms of degree. The formulas for a circle’s segment are as follows:

☯︎Formula To Calculate Area of a Segment of a Circle:-

✈︎Area of a Segment in RadiansA = (½) × r2 (θ – Sin θ)

✈︎Area of a Segment in DegreesA = (½) × r 2 × [(π/180) θ – sin θ]

Answer 2

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Area of a Segment of a Circle Formula

Formula To Calculate Area of a Segment of a Circle

Area of a Segment in Radians A = (½) × r2 (θ – Sin θ)

Area of a Segment in Degrees A = (½) × r 2 × [(π/180) θ – sin θ]

Segment of a Circle Definition

A segment of a circle can be defined as a region bounded by a chord and a corresponding arc lying between the chord’s endpoints. In other words, a circular segment is a region of a circle which is created by breaking apart from the rest of the circle through a secant or a chord. We can also define segments as the parts that are divided by the circle’s arc and connected through a chord by the endpoints of the arc. It is to be noted that the segments do not contain the center point.

Types of Segments in a Circle

According to the definition, the part of the circular region which is enclosed between a chord and corresponding arc is known as a segment of the circle. There are two classifications of segments in a circle, namely the major segment and the minor segment. The segment having a larger area is known as the major segment and the segment having a smaller area is known as the minor segment.

Formula To Calculate Area of a Segment of a Circle

Area of a Segment in Radians A = (½) × r2 (θ – Sin θ)

Area of a Segment in Degrees A = (½) × r 2 × [(π/180) θ – sin θ]

The area of ΔAOB can be calculated in two steps, As shown in fig. 2,

Calculate the height of ΔAOB i.e. OP using Pythagoras theorem as given below:

OP = √[r2–(AB/2)2] if the length of AB is given

or, OP = r cos (θ/2), if θ is given (in degrees)

Calculate the area of ∆AOB using the formula:

(A area ΔAOB) = ½ × base × height = ½ × AB × OP

Now, substituting the values in the area of segment formula, the area can be calculated.

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