Class 10 RD Sharma Solutions Chapter 15 Areas Related to Circles Exercise 15.1 | Set 1 (original) (raw)
Last Updated : 24 Sep, 2024
Chapter 15 of RD Sharma's Class 10 Mathematics textbook, titled "Areas Related to Circles," is a crucial component of geometry studies. This chapter builds upon students' existing knowledge of circles and introduces more advanced concepts related to calculating areas of circular regions and figures associated with circles.
It's an important topic that forms the foundation for more complex mathematical and real-world applications involving circular shapes.
Areas Related to Circles refers to the study of calculating the area of:
- **Circles: The region enclosed by a circular boundary.
- **Sectors: A "pie slice" shaped region of a circle, bounded by two radii and an arc.
- **Segments: The region of a circle bounded by a chord and an arc.
- **Annulus: The region between two concentric circles.
Combinations of circular and rectilinear figures: Such as areas of shaded regions involving circles and rectangles or triangles.
**Question 1. Find the circumference and area of a circle with a radius of 4.2 cm.
**Solution:
Radius = 4.2 cm
Circumference of a circle = 2πr
= 2 × (22/7) × 4.2
= 26.4 cm
Area of a circle = πr2
= (22/7) x 4.22
= (22/7) × 4.2 × 4.2
= 55.44 cm2
Therefore, circumference = 26.4 cm and area of the circle = 55.44 cm2
**Question 2. Find the circumference of a circle whose area is 301.84 cm 2 .
**Solution:
Area of circle = 301.84 cm2
Area of a Circle = πr2 = 301.84 cm2
(22/7) × r2 = 301.84
r2 = 96.04
r = √96.04 = 9.8cm
Radius = 9.8 cm.
Circumference of a circle = 2πr
= 2 × (22/7) × 9.8
= 61.6 cm
Therefore, the circumference of the circle = 61.6 cm.
**Question 3. Find the area of a circle whose circumference is 44 cm.
**Solution:
Circumference = 44 cm
2πr = 44 cm
2 × (22/7) × r = 44
r = 7 cm
Area of a Circle = πr2
= (22/7) × 7 × 7
= 154 cm2
Therefore, area of the Circle = 154 cm2
**Question 4. The circumference of a circle exceeds the diameter by 16.8 cm. Find the circumference of the circle.
**Solution:
Let the radius of the circle be r cm
Diameter (d) = 2r
Circumference of a circle (C) = 2πr
C = d + 16.8
2πr = 2r + 16.8
2πr - 2r = 16.8
2r (π - 1) = 16.8
2r (3.14 – 1) = 16.8
r = 3.92 cm
Radius = 3.92 cm
Circumference (C) = 2πr
C = 2 × 3.14 × 3.92
= 24.62 cm
Therefore, circumference of the circle = 24.64 cm.
**Question 5. A horse is tied to a pole with 28 m long string. Find the area where the horse can graze.
**Solution:
Length of the string = 28 m
Area the horse can graze is the area of the circle with a radius equal to the length of the string.
Area of a Circle = πr2
= (22/7) × 28 × 28
= 2464 m2
Therefore, the area where horse can graze = 2464 m2
**Question 6. A steel wire when bent in the form of a square encloses an area of 121 cm 2 . If the same wire is bent in the form of a circle, find the area of the circle.
**Solution:
Area of the square = a2
= 121 cm2
Area of the circle = πr2
121 cm2 = a2
Therefore, a = 11 cm
Perimeter of square = 4a
= 4 × 11 = 44 cm
Perimeter of the square = Circumference of the circle
Circumference = 2πr
44 = 2(22/7)r
r = 7 cm
Area of the Circle = πr2
= (22/7) × 7 × 7
= 154 cm2
Therefore, the area of the circle = 154 cm2.
**Question 7. The circumference of two circles are in the ratio of 2:3. Find the ratio of their areas.
**Solution:
Circumference of a circle (C) = 2πr
Circumference of first circle = 2πr1
Circumference of second circle = 2πr2.
2πr1 : 2πr2 = 2:3
Therefore,
r1: r2 = 2: 3
Area of circle 1 = (πr1)2
Area of circle 2 = (πr2)2
Ratio = 22:32
= 4/9
Therefore, ratio of areas = 4: 9.
**Question 8. The sum of the radii of two circles is 140 cm and the difference of their circumference is 88 cm. Find the diameters of the circles.
**Solution:
Sum of radii of two circles i.e., r1 + r2 = 140 cm … (i)
Difference of their circumference,
C1 – C2 = 88 cm
2πr1 – 2πr2 = 88 cm
2(22/7)(r1 – r2) = 88 cm
(r1 – r2) = 14 cm
r1 = r2 + 14…..(ii)
From (i) and (ii)
r2 + r2 + 14 = 140
2r2 = 140 – 14
2r2 = 126
r2 = 63 cm
r1 = 63 + 14 = 77 cm
Therefore,
Diameter of circle 1 = 2 x 77 = 154 cm
Diameter of circle 2 = 2 × 63 = 126 cm
**Question 9. Find the radius of a circle whose circumference is equal to the sum of the circumferences of two circles of radii 15cm and 18cm.
**Solution:
Radius of circle 1 = r1 = 15 cm
Radius of circle 2 = r2 = 18 cm
C1 = 2πr1 , C2 = 2πr2
C = C1 + C2
2πr = 2πr1 + 2πr2
r = r1 + r2
r = 15 + 18
r = 33 cm
Therefore, the radius of the circle = 33 cm
**Question 10. The radii of two circles are 8 cm and 6 cm respectively. Find the radius of the circle having its area equal to the sum of the areas of two circles.
**Solution:
Radii of the two circles are 6 cm and 8 cm
Area of circle with radius 8 cm = π (8)2
= 64π cm2
Area of circle with radius 6cm = π (6)2
= 36π cm2
Sum of areas = 64π + 36π = 100π cm2
Let the radius of the circle be r cm
Area of the circle = 100π cm2
πr2 = 100π
r= √100 = 10 cm
Therefore, the radius of the circle = 10 cm.
Summary
Exercise 15.1 in RD Sharma's Class 10 Areas Related to Circles chapter focuses on calculating the area and circumference of circles. It covers problems involving the relationship between radius, diameter, circumference, and area. Students are required to apply formulas, solve equations, and perform calculations involving π. The exercise also includes word problems that relate these concepts to real-world scenarios, helping students understand practical applications of circular measurements.