NCERT Solutions for Chapter 12 Areas Related to Circles Class 10 Maths
Book Solutions1
The radii of two circles are 19 cm and 9 cm respectively. Find the radius of the circle which has circumference equal to the sum of the circumferences of the two circles.
Answer
We have, r1 = 19 cm
r2 = 9 cm
∴ Circumference of circle-I = 2πr1 = 2π (19) cm
Circumference of circle-II = 2πr2 = 2π (9) cm
Sum of the circumferences of circle-I and circle-II = 2π (19) + 2π (9) = 2π (19 + 9) cm = 2π (28) cm
Let R be the radius of the circle−III.
∴ Circumference of circle-III = 2πR
According to the condition,
2πR = 2π (28)
Thus, the radius of the new circle = 28 cm
Exercise 12.1
Page Number 225
2
The radii of two circles are 8 cm and 6 cm respectively. Find the radius of the circle having area equal to the sum of the areas of the two circles.
Answer
Exercise 12.1
Page Number 225
3
Figure depicts an archery target marked with its five scoring regions from the centre outwards as Gold, Red, Blue, Black and White. The diameter of the region representing gold score is 21 cm and each of the other bands is 10.5 cm wide. Find the area of each of the five scoring regions.
Answer
Exercise 12.1
Page Number 225
4
The wheels of a car are of diameter 80 cm each. How many complete revolutions does each wheel make in 10 minutes when the car is travelling at a speed of 66 km per hour?
Answer
Diameter of a wheel = 80 cm∴ Radius of the wheel = 80/2 = 40 cm
∴ Circumference of the wheel
Exercise 12.1
Page Number 226
5
Tick the correct answer in the following and justify your choice: If the perimeter and the area of a circle are numerically equal, then the radius of the circle is
(A) 2 units
(B) π units
(C) 4 units
(D) 7 units.
(A) 2 units
(B) π units
(C) 4 units
(D) 7 units.
Answer
Exercise 12.1
Page Number 226
1
Find the area of a sector of a circle with radius 6 cm if angle of the sector is 60°.
Answer
Here,r = 6 cm
θ = 60°
Exercise 12.2
Page Number 230
2
Find the area of a quadrant of a circle whose circumference is 22 cm.
Answer
Exercise 12.2
Page Number 230
3
The length of the minute hand of a clock is 14 cm. Find the area swept by the minute hand in 5 minutes.
Answer
[Length of minute hand] = [radius of the circle]⇒ r = 14 cm
∵ Angle swept by the minute hand in 60 minutes = 360°
Exercise 12.2
Page Number 230
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 sector. (Use π = 3.14)
Answer
Exercise 12.2
Page Number 230
5
In a circle of radius 21 cm, an arc subtends an angle of 60° at the centre. Find:
(i) the length of the arc
(ii) area of the sector formed by the arc
(iii) area of the segment formed by the corresponding
(i) the length of the arc
(ii) area of the sector formed by the arc
(iii) area of the segment formed by the corresponding
Answer
Exercise 12.2
Page Number 230
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)
Answer
Here, radius (r) = 15 cmSector angle θ = 60°
∴ Area of the sector with θ = 60°
Exercise 12.2
Page Number 230
7
A chord of a circle of 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)
Answer
Exercise 12.2
Page Number 230
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)
(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)
Answer
Here, Length of the rope = 5 m∴ Radius of the circular region grazed by the horse = 5 m
(i) Area of the circular portion grazed
Exercise 12.2
Page Number 230
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 Fig. Find:
(i) the total length of the silver wire required.
(ii) the area of each sector of the brooch.
(i) the total length of the silver wire required.
(ii) the area of each sector of the brooch.
Answer
Diameter of the circle = 35 mm
Exercise 12.2
Page Number 230
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.
Answer
Exercise 12.2
Page Number 231
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.
Answer
Here, radius (r) = 25 cmSector angle (θ) = 115°
∴ Area cleaned by each sweep of the blades
Exercise 12.2
Page Number 231
12
To warn ships for underwater rocks, a lighthouse spreads a red coloured 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)
Answer
Here, Radius (r) = 16.5 kmSector angle (θ) = 80°
∴ Area of the sea surface over which the ships are warned
Exercise 12.2
Page Number 231
13
A round table cover has six equal designs as shown in Fig. If the radius of the cover is 28 cm, find the cost of making the designs at the rate of Rs 0.35 per cm2. (Use √3 = 1.7)
Answer
Now, area of 1 design
= Area of segment APB
= Area of sector − Area of Δ AOB ...(2)
In Δ AOB, ∠AOB = 60°, OA = OB = 28 cm
∴ ∠OAB = 60° and ∠OBA = 60°
⇒ Δ AOB is an equilateral triangle.
⇒ AB = AO = BO
⇒ AB = 28 cm
Draw OM ⊥ AB
∴ In right Δ AOM, we have
Exercise 12.2
Page Number 231
14
Tick the correct answer in the following :
Area of a sector of angle p (in degrees) of a circle with radius R is
(A) p/180 × 2πR
(B) p/180 × π R2
(C) p/360 × 2πR
(D) p/720 × 2πR2
Answer
Exercise 12.2
Page Number 231
1
Find the area of the shaded region in Fig. PQ = 24 cm, PR = 7 cm and O is the centre of the circle.
Answer
Since O is the centre of the circle,∴ QOR is a diameter.
⇒ ∠RPQ = 90° [Angle in a semi-circle]
Now, in right ∆ RPQ,
Exercise 12.3
Page Number 234
2
Find the area of the shaded region in figures, if radii of the two concentric circles with centre O are 7 cm and 14 cm respectively and angle AOC = 40°.
Answer
Exercise 12.3
Page Number 235
3
Find the area of the shaded region in Fig. 12.21, if ABCD is a square of side 14 cm and APD and BPC are semicircles.
Answer
Exercise 12.3
Page Number 235
4
Find the area of the shaded region in figure, where a circular arc of radius 6 cm has been drawn with vertex O of an equilateral triangle OAB of side 12 cm as centre.
Answer
Exercise 12.3
Page Number 235
5
From each corner of a square of side 4 cm a quadrant of a circle of radius 1 cm is cut and also a circle of diameter 2 cm is cut as shown in figure. Find the area of the remaining portion of the square.
Answer
Exercise 12.3
Page Number 235
6
In a circular table cover of radius 32 cm, a design is formed leaving an equilateral triangle ABC in the middle as shown in figure. Find the area of the design.
Answer
Exercise 12.3
Page Number 235
7
In figure, ABCD is a square of side 14 cm. With centres A, B, C and D, four circles are drawn such that each circle touch externally two of the remaining three circles. Find the area of the shaded region.
Answer
Exercise 12.3
Page Number 236
8
The figure depicts a racing track whose left and right ends are semicircular.
The distance between the two inner parallel line segments is 60 m and they are each 106 m long.
If the track is 10 m wide, find:
(i) the distance around the track along its inner edge
(ii) the area of the track
The distance between the two inner parallel line segments is 60 m and they are each 106 m long.
If the track is 10 m wide, find:
(i) the distance around the track along its inner edge
(ii) the area of the track
Answer
Exercise 12.3
Page Number 236
9
In the figure, AB and CD are two diameters of a circle (with centre O) perpendicular to each other and OD is the diameter of the smaller circle. If OA = 7 cm, find the area of the shaded region.
Answer
O is the centre of the circle.OA =7 cm
⇒ AB = 2 OA = 2 × 7 = 14 cm
OC = OA = 7 cm
∵ AB and CD are perpendicular to each other
⇒ OC ⊥ AB
Exercise 12.3
Page Number 236
10
The area of an equilateral triangle ABC is 17320.5 cm2. With each vertex of the triangle as centre, a circle is drawn with radius equal to half the length of the side of the triangle (see Fig.). Find the area of the shaded region.
(Use π = 3.14 and √3 = 1.73205).
(Use π = 3.14 and √3 = 1.73205).
Answer
Exercise 12.3
Page Number 236
11
On a square handkerchief, nine circular designs each of radius 7 cm are made (see figure). Find the area of the remaining portion of the handkerchief.
Answer
Exercise 12.3
Page Number 237
12
In the figure, OACB is a quadrant of a circle with centre O and radius 3.5 cm. If OD = 2 cm, find the area of the |
(i) quadrant OACB,
(ii) shaded region.
(i) quadrant OACB,
(ii) shaded region.
Answer
Exercise 12.3
Page Number 237
13
In the figure, a square OABC is inscribed in a quadrant OPBQ. If OA = 20 cm, find the area of the shaded region. (Use π = 3.14)
Answer
Exercise 12.3
Page Number 237
14
AB and CD are respectively areas of two concentric circles of radii 21 cm and 7 cm and centre O (see figure). If ∠AOB = 30°, find the area of the shaded region.
Answer
Exercise 12.3
Page Number 237
15
In the figure, ABPC is a quadrant of a circle of radius 14 cm and a semi-circle is drawn with BC as diameter. Find the area of the shaded region.
Answer
Exercise 12.3
Page Number 237
16
Calculate the area of the designed region in the figure, common between the two quadrants of circles of radius 8 cm each.
Answer
= 352/7 cm2
Exercise 12.3
Page Number 238