Class 10 Maths Areas Related to Circles Notes

11.1 Areas of Sector and Segment of a Circle

In this chapter, we study areas of different parts of a circle, specifically sectors and segments. These concepts are widely used in geometry, architecture, and design problems.

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Sector of a Circle

A sector is the portion of a circle enclosed by two radii and the arc between them. Think of a sector like a “slice of pizza” from the circle.

• Radius (r): The distance from the center of the circle to any point on the circle.
• Central Angle (θ): The angle formed at the center of the circle by the two radii.

Formula for the area of a sector:$$\text{Area of sector} = \frac{\theta}{360^\circ} \times \pi r^2$$Area of sector=360∘θ​×πr2

Where:

• $\theta$θ = central angle in degrees
• $r$r = radius of the circle

Example:
A circle has a radius of 7 cm. Find the area of a sector with a central angle of $60^\circ$60∘.$$\text{Area} = \frac{60}{360} \times \pi (7)^2 = \frac{1}{6} \times 49\pi = \frac{49\pi}{6} \approx 25.6 \, \text{cm}^2$$Area=36060​×π(7)2=61​×49π=649π​≈25.6cm2

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Segment of a Circle

A segment of a circle is the region bounded by a chord and the arc it subtends.

• The major segment is the larger part of the circle cut off by the chord.
• The minor segment is the smaller part of the circle cut off by the chord.

To find the area of a segment, we use:$$\text{Area of segment} = \text{Area of sector} – \text{Area of triangle formed by the radii and chord}$$Area of segment=Area of sector−Area of triangle formed by the radii and chord

Step-by-Step Example:
A circle has a radius of 7 cm, and a chord forms a sector with a central angle of $60^\circ$60∘. To find the area of the minor segment:

1. Area of sector $= \frac{60}{360} \times \pi \times 7^2 = \frac{49\pi}{6}$=36060​×π×72=649π​
2. Area of triangle (formed by the two radii and the chord) using formula for an equilateral triangle:

$$\text{Area of triangle} = \frac{1}{2} r^2 \sin \theta = \frac{1}{2} \cdot 7^2 \cdot \sin 60^\circ = \frac{49}{2} \cdot \frac{\sqrt{3}}{2} = \frac{49\sqrt{3}}{4}$$Area of triangle=21​r2sinθ=21​⋅72⋅sin60∘=249​⋅23​​=4493​​

3. Area of segment $= \text{Area of sector} – \text{Area of triangle}$=Area of sector−Area of triangle

$$\text{Area of segment} = \frac{49\pi}{6} – \frac{49\sqrt{3}}{4} \, \text{cm}^2$$Area of segment=649π​−4493​​cm2

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Key Points to Remember

1. A sector is like a “slice of the circle,” bounded by two radii and the arc.
2. A segment is the part cut off by a chord, which may be major or minor.
3. Formulas:
   • Area of sector: $\frac{\theta}{360} \pi r^2$360θ​πr2
   • Area of triangle inside sector: $\frac{1}{2} r^2 \sin \theta$21​r2sinθ
   • Area of segment: $\text{Area of sector} – \text{Area of triangle}$Area of sector−Area of triangle

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11.2 Summary

In this chapter, we learned:

• The difference between a sector and a segment of a circle.
• How to calculate the area of a sector using the radius and central angle.
• How to calculate the area of a segment by subtracting the area of the triangle from the area of the sector.
• Applications of these formulas in practical geometry problems.

By practicing various problems, students can quickly master areas related to sectors and segments for exams.

MCQs Based on the “Areas Related to Circles” Chapter:

1. The area of a sector of a circle is calculated using which formula?

a) $\pi r^2$πr2
b) $\frac{\theta}{360} \pi r^2$360θ​πr2
c) $\pi r$πr
d) $\frac{1}{2} r^2 \sin \theta$21​r2sinθ

Answer: b) $\frac{\theta}{360} \pi r^2$360θ​πr2

2. The area of a segment of a circle is obtained by:

a) Adding the area of the sector and triangle
b) Subtracting the area of the sector from the triangle
c) Subtracting the area of the triangle from the sector
d) Dividing the area of the sector by 2

Answer: c) Subtracting the area of the triangle from the sector

3. A circle has radius 7 cm. A sector has a central angle of 90∘90^\circ90∘. What is the area of the sector?

a) $\frac{49\pi}{4}$449π​ cm²
b) $\frac{49\pi}{2}$249π​ cm²
c) $\frac{49\pi}{6}$649π​ cm²
d) $\frac{7\pi}{2}$27π​ cm²

Answer: a) $\frac{49\pi}{4}$449π​ cm²

4. Which of the following represents a segment of a circle?

a) The area enclosed by two radii and an arc
b) The area enclosed by a chord and the arc
c) The entire area inside a circle
d) The circumference of the circle

Answer: b) The area enclosed by a chord and the arc

5. If the central angle of a sector is 60∘60^\circ60∘ and radius is 6 cm, the area of the sector is:

a) $6\pi$6π cm²
b) $12\pi$12π cm²
c) $\pi$π cm²
d) $36\pi$36π cm²

Answer: a) $6\pi$6π cm²