Gujarat Board Solutions Class 10 Maths Chapter 12 Areas Related to Circles

Gujarat Board Textbook Solutions Class 10 Maths Chapter 12 Areas Related to Circles

Ex 12.1

Question 1.
The radii of two circles are 19 cm and 9 cm respectively. Find the radius of the circle which has a circumference equal to the sum of the circumferences of the two circles.
Solution:
Let radii of two circles be r1 and r2, and R be the radius of third circle whose circumference is equal to the sum of circumference of two circle
r1 = 19 cm
r2 = 9 cm
C1 + C2 = C
2πr2 + 2πr2 = 2πR
⇒ 2π(r1 + r2) = 2πR
⇒ r1 + r1 = R
⇒ 19 + 9 = R
⇒ R = 28 cm

Question 2.
The radii of two circles are 8 cm and 6 cm respectively. Find the radius of the circle having an area equal to the sum of the areas of the two circles.
Solution:
Let r1 and r2 be the area of two circles and radius of the third circle be R.
r1 = 8 cm, r2 = 6 cm
A1 + A2 = A
⇒ πr21 + πr22 = πR2
⇒ π(r21 + r22) = πR2
⇒ r21 + r22 = R2
⇒ 82 + 62 = R2
⇒ R2 = 64 + 36 = 100
⇒ R = 10 cm

Question 3.
The figure depicts an archery target marked with its five scoring regions from the center outwards as Gold, Red, Blue, Black, and White. The diameter of the region representing the Gold score 21 cm and each of the other bands is 10.5 cm wide. Find the area of each of the five scoring regions.

Question 4.
The wheels of a car are of a diameter of 80 cm each. How many complete revolutions does each wheel make in 10 minutes? When the car is traveling at a speed of 66 km per hour?
Solution:

Question 5.
Tick the correct answer in the following and justify your choice. If the perimeter and 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
Solution:
Let the radius of the circle be r units. Then according to question
Area of circle = Perimeter of the circle
πr2 = 2πr
r = 2 units
Hence, the correct answer is (a) 2 units.

Ex 12.2

Question 1.
Find the area of a sector of a circle with radius 6 cm, if angle of the sector is 60°.
Solution:
Here ∠AOB = 60°
and radius r = 6 cm

Question 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)
Solution:
Here r = 15 cm
θ = 60°

Question 7.
A chord of circle of radius 12 cm subtends an angle of 120° at the centre. Find the area of the corresponding segments of the circle.
Solution:
Here r = 12 cm
and θ = 120°

Question 8.
A horse is tied to a leg 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 of the rope were 10 m long instead of 5 m. (Use π = 3.14)

Question 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:

Question 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.

Question 11.
A car has two wipers which do not overlap. Each wiper has a blade of length 25 cm sweeping through angle of 115°. Find the total area cleaned at each sweep of the blades.
Solution:
Here
r = 25 cm
θ = 115°
Total area cleaned at each sweep of the blades

Question 12.
Two 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)
Solution:
Here r = 16.5 km
and θ = 80°
The area of the sea over which the ships are warned

Question 13.
A round table cover has six equal designs as shown in figure. If the radius of the cover is 28 cm, find the cost of making the design at the rate of ₹ 0.35 per cm2. (Use √3 = 1.7)

Question 14.
Tick the correct answer in the following: Area of a sector of angle p (in degrees) of a circle with radius R is

Ex 12.3

Question 1.
Find the area of shaded region in the given figure. If PQ = 24 cm, PR = 7 cm, and O is the centre of the circle. (CBSE 2012)

Question 4.
Find the area of the shaded region in the given 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. (CBSE 2016)

Question 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.

Question 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 (Shaded region).

Question 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 shaded region.

Question 8.
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.

Question 9.
In 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. (CBSE 2000, 2010, 2013)

Question 10.
The area of an equilateral triangle ABC is 17320.5 cm2. With each vertex of the triangle as centre, a circle is dräwn with radius equal to half the length of side of the triangle (see figure). Find the area of shaded region. (Use π = 3.14 and √3 = 1.73205)

Question 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.

Question 12.
In figure OACB is a quadrants of circle with centre O and radius 3.5 cm. If OD = 2 cm, find the area of the
(i) quadrant OACB
(ii) shaded region.

Question 13.
In figure, a square OABC is inscribed in a quadrant OPBQ If OA = 20 cm, find the area of the shaded region.

Question 14.
AB and CD are respectively arcs of two concentric circles of radii 21 cm and 7 cm and centre O (see figure). If ∠AOB = 30°, find the area of shaded region.