Areas Related to Circles: Segment Area MCQs Quiz | Class 10

Understanding Areas Related to Circles: Segment Area

This section provides an educational overview of areas related to circles, with a special focus on the area of a segment. The concepts discussed here are crucial for Class X Mathematics, particularly within the Mensuration unit.

Key Concepts

1. Circle Basics

• Radius (r): The distance from the center of the circle to any point on its circumference.
• Chord: A line segment joining any two points on the circumference.
• Arc: A continuous piece of a circle’s circumference.
• Sector: The region of a circle enclosed by two radii and the arc between them.
• Segment: The region of a circle bounded by a chord and the arc subtended by the chord. It can be a minor segment (smaller part) or a major segment (larger part).

2. Area of a Sector

The area of a sector with radius ‘r’ and central angle ‘theta’ (in degrees) is given by:

Area of Sector = (theta / 360) * Pi * r^2

Where Pi is approximately 22/7 or 3.14.

3. Area of a Triangle

When calculating the area of a segment, we often need the area of the triangle formed by the two radii and the chord. For a triangle with two sides ‘r’ (radii) and the included angle ‘theta’, the area is:

Area of Triangle = (1/2) * r^2 * sin(theta)

It’s important to remember the values of sin for common angles:

4. Area of a Segment

The area of a minor segment is the difference between the area of the sector and the area of the triangle formed by the radii and the chord.

Area of Minor Segment = Area of Sector – Area of Triangle

Area of Minor Segment = (theta / 360) * Pi * r^2 – (1/2) * r^2 * sin(theta)

Special Cases for Central Angles

Case 1: Central Angle = 60 degrees

When the central angle is 60 degrees, the triangle formed by the two radii and the chord is an equilateral triangle (since two sides are radii and the angle between them is 60 degrees, the other two angles must also be 60 degrees). Its side length is ‘r’.

• Area of Sector = (60 / 360) * Pi * r^2 = (1/6) * Pi * r^2
• Area of Equilateral Triangle = (sqrt(3) / 4) * r^2
• Area of Segment = (1/6) * Pi * r^2 – (sqrt(3) / 4) * r^2

Case 2: Central Angle = 90 degrees

When the central angle is 90 degrees, the triangle formed is a right-angled isosceles triangle.

• Area of Sector = (90 / 360) * Pi * r^2 = (1/4) * Pi * r^2
• Area of Right Triangle = (1/2) * base * height = (1/2) * r * r = (1/2) * r^2
• Area of Segment = (1/4) * Pi * r^2 – (1/2) * r^2

Case 3: Central Angle = 120 degrees

When the central angle is 120 degrees, the triangle formed is an isosceles triangle.

• Area of Sector = (120 / 360) * Pi * r^2 = (1/3) * Pi * r^2
• Area of Triangle = (1/2) * r^2 * sin(120) = (1/2) * r^2 * (sqrt(3) / 2) = (sqrt(3) / 4) * r^2
• Area of Segment = (1/3) * Pi * r^2 – (sqrt(3) / 4) * r^2

Quick Revision Checklist

• Know the definitions of sector and segment.
• Remember the formula for the area of a sector.
• Master the formula for the area of a triangle given two sides and the included angle.
• Understand how to derive the area of a segment.
• Be familiar with the sine values for 60, 90, and 120 degrees.
• Practice calculations using Pi = 22/7 or 3.14.

Practice Questions (Without Options/Answers)

1. Find the area of a minor segment of a circle of radius 14 cm, when the angle of the corresponding sector is 60 degrees. (Use Pi = 22/7, sqrt(3) = 1.73)
2. A chord of a circle of radius 10 cm subtends a right angle at the centre. Find the area of the corresponding major segment. (Use Pi = 3.14)
3. In a circle with radius 35 cm, an arc subtends an angle of 120 degrees at the centre. Calculate the area of the minor segment. (Use Pi = 22/7, sqrt(3) = 1.73)
4. The area of a sector of a circle with radius 6 cm is 132/7 sq cm. Find the corresponding central angle. (Use Pi = 22/7)
5. What is the difference between the area of a sector and the area of the triangle formed by its radii and chord when the central angle is 90 degrees and radius is ‘r’?