Chapter 11 of NCERT Maths Class 10 delves into the concept of areas related to circles, focusing on calculating the areas of sectors, segments, and other shapes involving circles.
Key Concepts and Formulas:
- Sector: A portion of a circle enclosed by two radii and the corresponding arc.
- Segment: The region of a circle enclosed by an arc and the chord joining its endpoints.
- Area of a Circle: The area of a circle with radius r is given by πr².
- Area of a Sector: The area of a sector with central angle θ (in degrees) and radius r is given by (θ/360°) * πr².
- Area of a Segment: The area of a segment can be found by subtracting the area of the triangle formed by the chord and the radii from the area of the corresponding sector.
Applications:
- Finding the areas of various shapes involving circles, such as semi-circles, quarter-circles, and combinations of circles with other shapes.
- Solving real-world problems involving circular areas, like calculating the area of a pizza, a garden, or a circular pool.
- Understanding the relationship between the area of a circle and its radius.
In essence, this chapter provides a comprehensive understanding of the areas of different parts of circles, equipping students with the necessary tools to solve a variety of geometric problems.
Exercise 11.1
1. Find the area of a sector of a circle with radius 6 cm if angle of the sector is 60°.
Ans :
Formula for the area of a sector:
Area of sector = (θ/360°) * πr²
where:
- θ is the central angle
- r is the radius of the circle
Given:
- θ = 60°
- r = 6 cm
Substituting the values:
Area of sector
= (60°/360°) * π * 6²
= (1/6) * π * 36 = 6π cm²
Therefore, the area of the sector is 6π square centimeters.
2. Find the area of a quadrant of a circle whose circumference is 22 cm.
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3. The length of the minute hand of a clock is 14 cm. Find the area swept by the minute hand in 5 minutes.
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4. A chord of a circle of radius 10 cm subtends a right angle at the centre. Find the area of the corresponding:
(i) minor segment
(ii) major segment (Use π = 3.14)
Ans :
Step 1: Find the area of the sector
- Area of sector = (θ/360°) * πr²
- Here, θ = 90°, r = 10 cm, and π = 3.14
- Area of sector = (90/360) * 3.14 * 10²
- Area of sector = (1/4) * 314
- Area of sector = 78.5 cm²
Step 2: Find the area of the triangle formed by the chord and the radii
- Since the chord subtends a right angle at the center, the triangle formed is a right-angled triangle with each leg being the radius of the circle.
- Area of triangle = (1/2) * base * height
- Area of triangle = (1/2) * 10 * 10
- Area of triangle = 50 cm²
Step 3:
- Area of minor segment
- = Area of sector – Area of triangle
- Area of minor segment = 78.5 cm² – 50 cm²
- Area of minor segment = 28.5 cm²
Step 4: Find the area of the major segment
- Area of major segment = Area of circle – Area of minor segment
- Area of circle = πr²
- = 3.14 * 10²
- = 314 cm²
- Area of major segment = 314 cm² – 28.5 cm²
- Area of major segment = 285.5 cm²
Therefore, the area of the minor segment is 28.5 cm² and the area of the major segment is 285.5 cm².
5. In a circle of radius 21 cm, an arc subtends an angle of 60° at the centre. Find:
(i) length of the arc.
(ii) area of the sector formed by the arc.
(iii) area of the segment formed by the corresponding chord.
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6. A chord of a circle of radius 15 cm subtends an angle of 60° at the centre. Find the areas of the corresponding minor and major segments of the circle. (Use π = 3.14 and √3 = 1.73)
Ans :
Step 1: Find the area of the sector
- Area of sector = (θ/360°) * πr²
- Here, θ = 60°, r = 15 cm, π = 3.14
- Area of sector
- = (60°/360°) * 3.14 * 15²
- Area of sector = (1/6) * 3.14 * 225
- Area of sector = 117.75 cm²
Step 2: Find the area of the triangle formed by the chord and the radii
- Since the chord subtends a 60° angle at the center, the triangle formed is an equilateral triangle.
- In this case, a = 15 cm (radius of the circle)
- Area of triangle = (√3/4) * 15²
- Area of triangle = (225√3)/4 ≈ 97.31 cm²
Step 3: Find the areas of the minor and major segments
- Area of minor segment
- = Area of sector – Area of triangle
- Area of minor segment ≈ 117.75 cm² – 97.31 cm²
- Area of minor segment ≈ 20.44 cm²
- Area of major segment = Area of circle – Area of minor segment
- Area of circle = πr²
- = 3.14 * 15² = 706.5 cm²
- Area of major segment = 706.5 cm² – 20.44 cm²
- Area of major segment ≈ 686.06 cm²
Therefore, the area of the minor segment is approximately 20.44 cm² and the area of the major segment is approximately 686.06 cm².
7. A chord of a circle of the radius 12 cm subtends an angle of 120° at the centre. Find the area of the corresponding segment of the circle. (Use π = 3.14 and √3 = 1.73).
Ans :
Step 1: Find the area of the sector
- Area of sector = (θ/360°) * πr²
- Here, θ = 120°, r = 12 cm,
- and π = 3.14
- Area of sector = (120°/360°) * 3.14 * 12²
- Area of sector = (1/3) * 3.14 * 144
- Area of sector = 150.72 cm²
Step 2: Find the area of the triangle formed by the chord and the radii
- Since the chord subtends a 120° angle at the center, the triangle formed is an equilateral triangle with side length equal to the radius (12 cm).
- Area of triangle = (√3/4) * 12²
- Area of triangle = 36√3 cm²
- Using √3 = 1.73, Area of triangle ≈ 36 * 1.73 ≈ 62.28 cm²
Step 3:
- Area of segment ≈ 150.72 cm² – 62.28 cm²
- Area of segment ≈ 88.44 cm²
Therefore, the area of the corresponding segment of the circle is approximately 88.44 square centimeters.
8. A horse is tied to a peg at one corner of a square shaped grass field of side 15 m by means of a 5 m long rope (see figure). Find:
(i) the area of that part of the field in which the horse can graze.
(ii) the increase in the grazing area if the rope were 10 m long instead of 5 m. (Use π = 3.14)
Ans :
(i) Area of Grazed Land with 5m Rope
- The horse can graze in a quarter-circle with a radius of 5 meters.
- Area of a quarter-circle = (1/4) * πr²
- Substituting r = 5: Area = (1/4) * π * 5²
- Area = 25π/4 ≈ 19.625 square meters
(ii) Area of Grazed Land with 10m Rope
- The horse can now graze in a quarter-circle with a radius of 10 meters.
- Area = (1/4) * π * 10²
- Area = 100π/4 = 25π ≈ 78.5 square meters
Increase in Grazing Area
- Increase = Area with 10m rope – Area with 5m rope
- Increase = 78.5 – 19.625 ≈ 58.875 square meters
9. A brooch is made with silver wire in the form of a circle with diameter 35 mm. The wire is also used in making 5 diameters which divide the circle into 10 equal sectors as shown in figure.
Find:
(i) the total length of the silver wire required.
(ii) the area of each sector of the brooch.
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10. An umbrella has 8 ribs which are equally spaced (see figure). Assuming umbrella to be a flat circle of radius / 45 cm, find the area between the two consecutive ribs of the umbrella.
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11. A car has two wipers which do not overlap.
Each wiper has a blade of length 25 cm sweeping through an angle of 115°. Find the total area cleaned at each sweep of the blades.
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12. To warn ships for underwater rocks, a lighthouse spreads a red colored light over a sector of angle 80° to a distance of 16.5 km. Find the area of the sea over which the ships are warned. (Use π = 3.14)
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The lighthouse spreads the red light over a sector of 80° with a radius of 16.5 km.
The area of a sector is given by:
Area = (θ/360°) * πr²
where θ is the central angle of the sector (80°), and r is the radius (16.5 km).
Substituting the values:
Area = (80°/360°) * π * (16.5)² Area = (2/9) * π * (272.25) Area ≈ 189.97 km²
Therefore, the area of the sea over which the ships are warned is approximately 189.97 square kilometers.
13. A round table cover has six equal designs as shown in the figure. If the radius of the cover is 28 cm, find the cost of making the designs at the rate of ₹0.35 per cm². (Use √3= 1.7)
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14. Tick the correct answer in the following: Area of a sector of angle p (in degrees) of a circle with radius R is
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The correct answer is (C) p/360 × πR².
