12.Areas related to the circles

Exercise 12.1

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:

The radius of the 1st circle = 19 cm (given)

∴ Circumference of the 1st circle = 2π×19 = 38π cm

The radius of the 2nd circle = 9 cm (given)

∴ Circumference of the 2nd circle = 2π×9 = 18π cm

So,

The sum of the circumference of two circles = 38π+18π = 56π cm

Now, let the radius of the 3rd circle = R

∴ The circumference of the 3rd circle = 2πR

It is given that sum of the circumference of two circles = circumference of the 3rd circle

Hence, 56π = 2πR

Or, R = 28 cm.

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.

Solution:

Radius of 1st circle = 8 cm (given)

∴ Area of 1st circle = π(8)2 = 64π

Radius of 2nd circle = 6 cm (given)

∴ Area of 2nd circle = π(6)2 = 36π

So,

The sum of 1st and 2nd circle will be = 64π+36π = 100π

Now, assume that the radius of 3rd circle = R

∴ Area of the circle 3rd circle = πR2

It is given that the area of the circle 3rd circle = Area of 1st circle + Area of 2nd circle

Or, πR2 = 100πcm2

R2 = 100cm2

So, R = 10cm

3. Fig. 12.3 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.

Ncert solution class 10 chapter 12-1

Solution:

The radius of 1st circle, r1 = 21/2 cm (as diameter D is given as 21 cm)

So, area of gold region = π r12 = π(10.5)2 = 346.5 cm2

Now, it is given that each of the other bands is 10.5 cm wide,

So, the radius of 2nd circle, r2 = 10.5cm+10.5cm = 21 cm

Thus,

∴ Area of red region = Area of 2nd circle − Area of gold region = (πr22−346.5) cm2

= (π(21)2 − 346.5) cm2

= 1386 − 346.5

= 1039.5 cm2

Similarly,

The radius of 3rd circle, r3 = 21 cm+10.5 cm = 31.5 cm

The radius of 4th circle, r4 = 31.5 cm+10.5 cm = 42 cm

The Radius of 5th circle, r5 = 42 cm+10.5 cm = 52.5 cm

For the area of nth region,

A = Area of circle n – Area of circle (n-1)

∴ Area of blue region (n=3) = Area of third circle – Area of second circle

= π(31.5)2 – 1386 cm2

= 3118.5 – 1386 cm2

= 1732.5 cm2

∴ Area of black region (n=4) = Area of fourth circle – Area of third circle

= π(42)2 – 1386 cm2

= 5544 – 3118.5 cm2

= 2425.5 cm2

∴ Area of white region (n=5) = Area of fifth circle – Area of fourth circle

= π(52.5)2 – 5544 cm2

= 8662.5 – 5544 cm2

= 3118.5 cm2

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?

Solution:

The radius of car’s wheel = 80/2 = 40 cm (as D = 80 cm)

So, the circumference of wheels = 2πr

= 80 π cm

Now, in one revolution, the distance covered = circumference of the wheel = 80 π cm

It is given that the distance covered by the car in 1 hr = 66km

Converting km into cm we get,

Distance covered by the car in 1hr = (66×105) cm

In 10 minutes, the distance covered will be = (66×105×10)/60 = 1100000 cm/s

∴ Distance covered by car = 11×105 cm

Now, the no. of revolutions of the wheels = (Distance covered by the car/Circumference of the wheels)

=( 11×105)/80 π = 4375.

5. Tick the correct Solution: 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

Solution:

Since the perimeter of the circle = area of the circle,

2πr = πr2

Or, r = 2

So, option (A) is correct i.e. the radius of the circle is 2 units.

exercise 12.2

Q1.Find the area of a sector of a circle with a radius 6 cm if the angle of the sector is 60°[use π = 22/7]

Ans.

Q1 ex 12.2 area related to circle

We are given the radius of the circle,r = 6 cm, angle of the sector,θ = 60°

$\boldsymbol{Area =\frac{\Theta }{360^{\circ}}\times \pi r^{2}}$

$\boldsymbol{ =\frac{60^{\circ} }{360^{\circ}}\times \frac{22}{7}\times 6^{2}}$

$\boldsymbol{ =\frac{60^{\circ} }{360^{\circ}}\times \frac{22}{7}\times 6\times 6}$

$\boldsymbol{=\frac{1}{6}\times \frac{22}{7}\times 6\times 6=\frac{132}{7}}$

Therefore area of the sector = 132/7 cm²

Q2.Find the area of a quadrant of a circle whose circumference is 22 cm.[use π = 22/7].

Circumference of the circle is = 2πr, where is the radius of the circle

According to question

2πr = 22

2×(22/7)r = 22

r = 7/2

r = 3.5

The quadrant of the circle is one-fourth of the circle, so it subtends the angle at the centre = 360/4 = 90°

Quadrant is the sector of the circle which subtends the angle 90° at the centre

$\boldsymbol{Area =\frac{\Theta }{360^{\circ}}\times \pi r^{2}}$

$\boldsymbol{=\frac{90}{360}\times \frac{22}{7}\times 3.5\times 3.5}$

$\boldsymbol{=\frac{1}{4}\times 22\times 0.5\times 3.5}$

$\boldsymbol{=5.5\times 0.5\times 3.5}$

= 9.625

Hence the area of the circle is = 9.625 sq.cm

Q3. The length of the minute hand of a clock is 14 cm. Find the area swept by the minute hand in 5 minutes. [use π = 22/7]

Q3 EX 12.2 area related to area

In 60 minutes, minute hand creates = 360°

In 1 minute minute hand creates = 360°/60 = 6°

In 5 minute, minute hand creates = 6°×5= 30°

$\boldsymbol{Area =\frac{\Theta }{360^{\circ}}\times \pi r^{2}}$

Area swept in 5 minutes by the minute hand

$\boldsymbol{=\frac{30}{360}\times \frac{22}{7}\times 14\times 14}$

$\boldsymbol{=\frac{1}{12}\times 22\times 2\times 14}$

$\boldsymbol{=\frac{1}{3}\times 11\times 2\times 7}$

$\boldsymbol{=\frac{154}{3}}$

Therefore area swept in 5 minutes by the minute hand = 154/3 cm²

Q4. 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]

Ans.

q4 ex 12.2 area related to circle

(i) Ans. We are given a chord PQ subtending 90° at the centre

The area of the minor segment PQS= Area of the corresponding sector OPSQ-Area of the triangle POQ.

Area of sector is given by

$\boldsymbol{Area =\frac{\Theta }{360^{\circ}}\times \pi r^{2}}$

Where θ= angle subtended by the chord, radius,r = 10 cm

Area of minor sector OPSQ

$\boldsymbol{Area\: of \: OPSQ=\frac{90}{360}\times 3.14\times 10\times 10}$

$\boldsymbol{=\frac{1}{4}\times 3.14\times 10\times 10}$

$\boldsymbol{=\frac{1}{2}\times 3.14\times 5\times 10}$

=3.14 × 5 × 5

= 78.5

Area of sector OPSQ = 78.5 cm²

Area of the triangle POQ

$\boldsymbol{=\frac{1}{2}\times Base\times Altitude}$

$\boldsymbol{=\frac{1}{2}\times 10\times 10=50}$

Area of triangle POQ is = 50 cm²

Hence area of the minor segment PQS

= 78.5 – 50 = 28.5 cm²

(ii) Angle subtended by the major arc PRQ at the centre = 360-90 = 270°

Area of major sector

$\boldsymbol{=\frac{270}{360}\times 3.14\times 10\times 10}$

$\boldsymbol{=\frac{3}{4}\times 3.14\times 10\times 10}$

$\boldsymbol{=\frac{3}{2}\times 3.14\times 5\times 10}$

$\boldsymbol{= 3\times 3.14\times 5\times 5}$

= 235.5

Hence area of major sector is = 235.5 cm²

Q5. 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 chord

$\boldsymbol{\left [use\: \pi =\frac{22}{7} \right ]}$

Ans.

Q5 ex 12.2 area related to circle

We are given radius, r= 21 cm, angle subtended by the arc ACB ,θ= 60°

(i)Let the length of the arc = l

$\boldsymbol{l=\frac{\Theta }{360}\times \boldsymbol{2}\pi r}$

$\boldsymbol{=\frac{60 }{360}\times \boldsymbol{2}\times \frac{22}{7}\times 21}$

$\boldsymbol{=\frac{1 }{6}\times \boldsymbol{2}\times \frac{22}{7}\times 21}$

$\boldsymbol{=\frac{1 }{3}\times \boldsymbol{2}\times \frac{11}{7}\times 21}$

= 22

Hence the length of the arc is = 22 cm

(ii) Area of the sector formed by the arc ,AOBC

$\boldsymbol{Area =\frac{\Theta }{360^{\circ}}\times \pi r^{2}}$

$\boldsymbol{=\frac{60}{360}\times \frac{22}{7}\times 21\times 21}$

$\boldsymbol{=\frac{1}{6}\times \frac{22}{7}\times 21\times 21}$

$\boldsymbol{=\frac{1}{3}\times \frac{11}{7}\times 21\times 21}$

= 11× 21 = 231

Area of the sector formed by the arc i.e arAOBC = 231 sq.cm

(iii) Area of the segment(ACB) formed by the corresponding chord

= Area of the sector AOBC – Area of the triangle AOB

In triangle AOB

OA = OB (radii of the circle)

∠OAB = ∠OBA (angles opposite to equal sides of the triangle)

∠OAB + ∠OBA + ∠AOB = 180°

2∠OAB + 60° = 180°

∠OAB = 60°

∠OAB =∠OBA =∠AOB = 60°

Therefore triangle AOB is an equilateral triangle

The area of equilateral triangle with sides ‘a’

$\boldsymbol{=\frac{a^{2}}{4}\sqrt{3}}$

a = 21 cm

Area of ΔAOB

$\boldsymbol{=\frac{21^{2}}{4}\sqrt{3}}$

$\boldsymbol{=\frac{{441}}{4}\sqrt{3}\: cm^{2}}$

Hence area of the segment(ACB) formed by the corresponding chord

$\boldsymbol{=(231-\frac{{441}}{4}\sqrt{3})\: cm^{2}}$

Q6. A chord of a circle of radius 15 cm subtends an angle of 60° at the centre. Find the area of the corresponding minor and major segments of the circle.[use π = 3.14 and √3 = 1.73]

Ans.

Q6 ex 12.2 -area related to circle

We are given the radius of the circle ,r = 15 cm, angle subtended by the radii of the circle,θ = 60°

Area of the corresponding segment (minor segment) = Area of the corresponding sector – Area of the corresponding triangle

Area of the corresponding sector(AOBC)

$\boldsymbol{Area =\frac{\Theta }{360^{\circ}}\times \pi r^{2}}$

$\boldsymbol{=\frac{60}{360}\times 3.14\times 15\times 15}$

$\boldsymbol{=\frac{1}{6}\times 3.14\times 15\times 15}$

$\boldsymbol{=\frac{1}{2}\times 3.14\times 5\times 15}$

$\boldsymbol{= 3.14\times 5\times 7.5}$

=117.5

Area of sector AOBC is 117.5 cm²

Triangle AOB is an equilateral Δ(see Q.5)

$\boldsymbol{=\frac{a^{2}}{4}\sqrt{3}}$

$\boldsymbol{=\frac{15\times 15}{4}\times 1.73}$

= 0.4325 × 225 =97.3125 cm²

Area of the corresponding minor segment (AOBC)

= 117.5 – 97.3125 = 20.1875 cm²

Area of major segment = Area of circle – Area of minor segment

= πr² – 20.1875

= 3.14 × 15× 15 -20.1875

= 706.25 -20.1875

= 686.0625 cm²

Hence the area of minor and major segments of the circle are 20.1875 cm² and 686.0625 cm² respectively.

Hence the radius of the circle is 2 units, so the answer is (i) 2 units

Q7.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.We are given a chord AB which subtends ∠AOB = 120° at the centre

Drawing OC⊥ AB,since ΔAOB is an isoscell triangle therefore OC will bisect the ∠AOB

∴∠BOC = 60°

sin60° = BC/BO =BC/12

Q7 Exercise 12.2 area related to circle

BC/12 = √3/2

BC = 12√3/2 = 6√3 cm

Since OC⊥ AB ,threfore OC bisects the chord AB

AB = 2×6√3 = 12√3 cm

cos60° = OC/BO =OC/12

OC/12 = 1/2

OC = 12/2 = 6 cm

arΔAOB = (1/2) AB×OC = (1/2)×12√3×6 = 36√3 =36×1.73 =62.28 cm²

Area of the sector AOBD =(θ/360°)πr² =(120°/360°)×3.14×12×12 =150.72 cm²

Area of the segment ADB = Area of the sector AOBD-arΔAOB =150.72-62.28 =88.44 cm

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 Fig. 12.11). 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)

Solution:

As the horse is tied at one end of a square field, it will graze only a quarter (i.e. sector with θ = 90°) of the field with radius 5 m.

Here, the length of rope will be the radius of the circle i.e. r = 5 m

It is also known that the side of square field = 15 m

(i) Area of circle = πr2 = 22/7 × 52 = 78.5 m2

Now, the area of the part of the field where the horse can graze = ¼ (the area of the circle) = 78.5/4 = 19.625 m2

(ii) If the rope is increased to 10 m,

Area of circle will be = πr2 =22/7×102 = 314 m2

Now, the area of the part of the field where the horse can graze = ¼ (the area of the circle)

= 314/4 = 78.5 m2

∴ Increase in the grazing area = 78.5 m2 – 19.625 m2 = 58.875 m2

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. 12.12. Find:

(i) the total length of the silver wire required.

(ii) the area of each sector of the brooch.

Ncert solution class 10 chapter 12-7

Solution:

Diameter (D) = 35 mm

Total number of diameters to be considered= 5

Now, the total length of 5 diameters that would be required = 35×5 = 175

Circumference of the circle = 2πr

Or, C = πD = 22/7×35 = 110

Area of the circle = πr2

Or, A = (22/7)×(35/2)2 = 1925/2 mm2

(i) Total length of silver wire required = Circumference of the circle + Length of 5 diameter

= 110+175 = 285 mm

(ii) Total Number of sectors in the brooch = 10

So, the area of each sector = total area of the circle/number of sectors

∴ Area of each sector = (1925/2)×1/10 = 385/4 mm2

10. An umbrella has 8 ribs which are equally spaced (see Fig. 12.13). Assuming umbrella to be a flat circle of radius 45 cm, find the area between the two consecutive ribs of the umbrella.

Ncert solution class 10 chapter 12-8

Solution:

The radius (r) of the umbrella when flat = 45 cm

So, the area of the circle (A) = πr2 = (22/7)×(45)2 =6364.29 cm2

Total number of ribs (n) = 8

∴ The area between the two consecutive ribs of the umbrella = A/n

6364.29/8 cm2

Or, The area between the two consecutive ribs of the umbrella = 795.5 cm2

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.

Solution:

Given,

Radius (r) = 25 cm

Sector angle (θ) = 115°

Since there are 2 blades,

The total area of the sector made by wiper = 2×(θ/360°)×π r2

= 2×(115/360)×(22/7)×252

= 2×158125/252 cm2

= 158125/126 = 1254.96 cm2

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)

Solution:

Let O bet the position of Lighthouse.

Ncert solution class 10 chapter 12-9

Here the radius will be the distance over which light spreads.

Given, radius (r) = 16.5 km

Sector angle (θ) = 80°

Now, the total area of the sea over which the ships are warned = Area made by the sector

Or, Area of sector = (θ/360°)×πr2

= (80°/360°)×πr2 km2

= 189.97 km2

13. A round table cover has six equal designs as shown in Fig. 12.14. 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)

Solution:

Total number of equal designs = 6

AOB= 360°/6 = 60°

Radius of the cover = 28 cm

Cost of making design = ₹ 0.35 per cm2

Since the two arms of the triangle are the radii of the circle and thus are equal, and one angle is 60°, ΔAOB is an equilateral triangle. So, its area will be (√3/4)×a2 sq. units

Here, a = OA

∴ Area of equilateral ΔAOB = (√3/4)×282 = 333.2 cm2

Area of sector ACB = (60°/360°)×πr2 cm2

= 410.66 cm2

So, area of a single design = area of sector ACB – area of ΔAOB

= 410.66 cm2 – 333.2 cm2 = 77.46 cm2

∴ Area of 6 designs = 6×77.46 cm2 = 464.76 cm2

So, total cost of making design = 464.76 cm2 ×Rs.0.35 per cm2

= Rs. 162.66

14. Tick the correct solution 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

Solution:

The area of a sector = (θ/360°)×πr2

Given, θ = p

So, area of sector = p/360×πR2

Multiplying and dividing by 2 simultaneously,

= (p/360)×2/2×πR2

= (2p/720)×2πR2

So, option (D) is correct.

Exercise 12.3

1. Find the area of the shaded region in Fig. 12.19, if PQ = 24 cm, PR = 7 cm and O is the centre of the circle.

Ncert solution class 10 chapter 12-12

Solution:

Here, P is in the semi-circle and so,

P = 90°

So, it can be concluded that QR is hypotenuse of the circle and is equal to the diameter of the circle.

∴ QR = D

Using Pythagorean theorem,

QR2 = PR2+PQ2

Or, QR2 = 72+242

QR= 25 cm = Diameter

Hence, the radius of the circle = 25/2 cm

Now, the area of the semicircle = (πR2)/2

= (22/7)×(25/2)×(25/2)/2 cm2

= 13750/56 cm2 = 245.54 cm2

Also, area of the ΔPQR = ½×PR×PQ

=(½)×7×24 cm2

= 84 cm2

Hence, the area of the shaded region = 245.54 cm2-84 cm2

= 161.54 cm2

2. Find the area of the shaded region in Fig. 12.20, if radii of the two concentric circles with centre O are 7 cm and 14 cm, respectively and AOC = 40°.

Solution:

Ncert solution class 10 chapter 12-13

Given,

Angle made by sector = 40°,

Radius the inner circle = r = 7 cm, and

Radius of the outer circle = R = 14 cm

We know,

Area of the sector = (θ/360°)×πr2

So, Area of OAC = (40°/360°)×πr2 cm2

= 68.44 cm2

Area of the sector OBD = (40°/360°)×πr2 cm2

= (1/9)×(22/7)×72 = 17.11 cm2

Now, area of the shaded region ABDC = Area of OAC – Area of the OBD

= 68.44 cm2 – 17.11 cm2 = 51.33 cm2

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.

Ncert solution class 10 chapter 12-14

Solution:

Side of the square ABCD (as given) = 14 cm

So, Area of ABCD = a2

= 14×14 cm2 = 196 cm2

We know that the side of the square = diameter of the circle = 14 cm

So, side of the square = diameter of the semicircle = 14 cm

∴ Radius of the semicircle = 7 cm

Now, area of the semicircle = (πR2)/2

= (22/7×7×7)/2 cm2

= 77 cm2

∴ Area of two semicircles = 2×77 cm2 = 154 cm2

Hence, area of the shaded region = Area of the Square – Area of two semicircles

= 196 cm2 -154 cm2

= 42 cm2

4. Find the area of the shaded region in Fig. 12.22, 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.

Ncert solution class 10 chapter 12-15

Solution:

It is given that OAB is an equilateral triangle having each angle as 60°

Area of the sector is common in both.

Radius of the circle = 6 cm.

Side of the triangle = 12 cm.

Area of the equilateral triangle = (√3/4) (OA)2= (√3/4)×122 = 36√3 cm2

Area of the circle = πR2 = (22/7)×62 = 792/7 cm2

Area of the sector making angle 60° = (60°/360°) ×πr2 cm2

= (1/6)×(22/7)× 62 cm2 = 132/7 cm2

Area of the shaded region = Area of the equilateral triangle + Area of the circle – Area of the sector

= 36√3 cm2 +792/7 cm2-132/7 cm2

= (36√3+660/7) cm2

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 Fig. 12.23. Find the area of the remaining portion of the square.

NCERT Solutions for Class 10 Maths Chapter 12 Exercise 12.3 Question Number 5

Solution:

Side of the square = 4 cm

Radius of the circle = 1 cm

Four quadrant of a circle are cut from corner and one circle of radius are cut from middle.

Area of square = (side)2= 42 = 16 cm2

Area of the quadrant = (πR2)/4 cm2 = (22/7)×(12)/4 = 11/14 cm2

∴ Total area of the 4 quadrants = 4 ×(11/14) cm2 = 22/7 cm2

Area of the circle = πR2 cm2 = (22/7×12) = 22/7 cm2

Area of the shaded region = Area of square – (Area of the 4 quadrants + Area of the circle)

= 16 cm2-(22/7) cm2-(22/7) cm2

= 68/7 cm2

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 Fig. 12.24. Find the area of the design.

Ncert solution class 10 chapter 12-16

Solution:

Radius of the circle = 32 cm

Draw a median AD of the triangle passing through the centre of the circle.

⇒ BD = AB/2

Since, AD is the median of the triangle

∴ AO = Radius of the circle = (2/3) AD

⇒ (2/3)AD = 32 cm

⇒ AD = 48 cm

In ΔADB,

Ncert solution class 10 chapter 12-17

By Pythagoras theorem,

AB2 = AD2 +BD2

⇒ AB2 = 482+(AB/2)2

⇒ AB2 = 2304+AB2/4

⇒ 3/4 (AB2)= 2304

⇒ AB2 = 3072

⇒ AB= 32√3 cm

Area of ΔADB = √3/4 ×(32√3)2 cm2 = 768√3 cm2

Area of circle = πR2 = (22/7)×32×32 = 22528/7 cm2

Area of the design = Area of circle – Area of ΔADB

= (22528/7 – 768√3) cm2

7. In Fig. 12.25, 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.

Ncert solution class 10 chapter 12-18

Solution:

Side of square = 14 cm

Four quadrants are included in the four sides of the square.

∴ Radius of the circles = 14/2 cm = 7 cm

Area of the square ABCD = 142 = 196 cm2

Area of the quadrant = (πR2)/4 cm2 = (22/7) ×72/4 cm2

= 77/2 cm2

Total area of the quadrant = 4×77/2 cm2 = 154cm2

Area of the shaded region = Area of the square ABCD – Area of the quadrant

= 196 cm2 – 154 cm2

= 42 cm2

8. Fig. 12.26 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.

Ncert solution class 10 chapter 12-19

Solution:

Width of the track = 10 m

Distance between two parallel lines = 60 m

Length of parallel tracks = 106 m

Ncert solution class 10 chapter 12-20

DE = CF = 60 m

Radius of inner semicircle, r = OD = O’C

= 60/2 m = 30 m

Radius of outer semicircle, R = OA = O’B

= 30+10 m = 40 m

Also, AB = CD = EF = GH = 106 m

Distance around the track along its inner edge = CD+EF+2×(Circumference of inner semicircle)

= 106+106+(2×πr) m = 212+(2×22/7×30) m

= 212+1320/7 m = 2804/7 m

Area of the track = Area of ABCD + Area EFGH + 2 × (area of outer semicircle) – 2 × (area of inner semicircle)

= (AB×CD)+(EF×GH)+2×(πr2/2) -2×(πR2/2) m2

= (106×10)+(106×10)+2×π/2(r2-R2) m2

= 2120+22/7×70×10 m2

= 4320 m2

9. In Fig. 12.27, 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.

Ncert solution class 10 chapter 12-21

Solution:

Radius of larger circle, R = 7 cm

Radius of smaller circle, r = 7/2 cm

Height of ΔBCA = OC = 7 cm

Base of ΔBCA = AB = 14 cm

Area of ΔBCA = 1/2 × AB × OC = (½)×7×14 = 49 cm2

Area of larger circle = πR2 = (22/7)×72 = 154 cm2

Area of larger semicircle = 154/2 cm2 = 77 cm2

Area of smaller circle = πr2 = (22/7)×(7/2)×(7/2) = 77/2 cm2

Area of the shaded region = Area of larger circle – Area of triangle – Area of larger semicircle + Area of smaller circle

Area of the shaded region = (154-49-77+77/2) cm2

= 133/2 cm2 = 66.5 cm2

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. 12.28). Find the area of the shaded region. (Use π = 3.14 and √3 = 1.73205)

Ncert solution class 10 chapter 12-22

Solution:

ABC is an equilateral triangle.

∴ ∠ A = ∠ B = ∠ C = 60°

There are three sectors each making 60°.

Area of ΔABC = 17320.5 cm2

⇒ √3/4 ×(side)2 = 17320.5

⇒ (side)2 =17320.5×4/1.73205

⇒ (side)2 = 4×104

⇒ side = 200 cm

Radius of the circles = 200/2 cm = 100 cm

Area of the sector = (60°/360°)×π r2 cm2

= 1/6×3.14×(100)2 cm2

= 15700/3cm2

Area of 3 sectors = 3×15700/3 = 15700 cm2

Thus, area of the shaded region = Area of equilateral triangle ABC – Area of 3 sectors

= 17320.5-15700 cm2 = 1620.5 cm2

11. On a square handkerchief, nine circular designs each of radius 7 cm are made (see Fig. 12.29). Find the area of the remaining portion of the handkerchief.

Solution:

Number of circular designs = 9

Radius of the circular design = 7 cm

There are three circles in one side of square handkerchief.

∴ Side of the square = 3×diameter of circle = 3×14 = 42 cm

Area of the square = 42×42 cm2 = 1764 cm2

Area of the circle = π r2 = (22/7)×7×7 = 154 cm2

Total area of the design = 9×154 = 1386 cm2

Area of the remaining portion of the handkerchief = Area of the square – Total area of the design = 1764 – 1386 = 378 cm2

12. In Fig. 12.30, 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.

Ncert solution class 10 chapter 12-24

Solution:

Radius of the quadrant = 3.5 cm = 7/2 cm

(i) Area of quadrant OACB = (πR2)/4 cm2

= (22/7)×(7/2)×(7/2)/4 cm2

= 77/8 cm2

(ii) Area of triangle BOD = (½)×(7/2)×2 cm2

= 7/2 cm2

Area of shaded region = Area of quadrant – Area of triangle BOD

= (77/8)-(7/2) cm2 = 49/8 cm2

= 6.125 cm2

13. In Fig. 12.31, a square OABC is inscribed in a quadrant OPBQ. If OA = 20 cm, find the area of the shaded region. (Use π = 3.14)

Solution:

Side of square = OA = AB = 20 cm

Radius of the quadrant = OB

OAB is right angled triangle

By Pythagoras theorem in ΔOAB,

OB2 = AB2+OA2

⇒ OB2 = 202 +202

⇒ OB2 = 400+400

⇒ OB2 = 800

⇒ OB= 20√2 cm

Area of the quadrant = (πR2)/4 cm2 = (3.14/4)×(20√2)2 cm2 = 628cm2

Area of the square = 20×20 = 400 cm2

Area of the shaded region = Area of the quadrant – Area of the square

= 628-400 cm2 = 228cm2

14. AB and CD are respectively arcs of two concentric circles of radii 21 cm and 7 cm and centre O (see Fig. 12.32). If ∠AOB = 30°, find the area of the shaded region.

Ncert solution class 10 chapter 12-26

Solution:

Radius of the larger circle, R = 21 cm

Radius of the smaller circle, r = 7 cm

Angle made by sectors of both concentric circles = 30°

Area of the larger sector = (30°/360°)×πR2 cm2

= (1/12)×(22/7)×212 cm2

= 231/2cm2

Area of the smaller circle = (30°/360°)×πr2 cm2

= 1/12×22/7×72 cm2

=77/6 cm2

Area of the shaded region = (231/2) – (77/6) cm2

= 616/6 cm2 = 308/3cm2

15. In Fig. 12.33, ABC is a quadrant of a circle of radius 14 cm and a semicircle is drawn with BC as diameter. Find the area of the shaded region.

Ncert solution class 10 chapter 12-27

Solution:

Radius of the quadrant ABC of circle = 14 cm

AB = AC = 14 cm

BC is diameter of semicircle.

ABC is right angled triangle.

By Pythagoras theorem in ΔABC,

BC2 = AB2 +AC2

⇒ BC2 = 142 +142

⇒ BC = 14√2 cm

Radius of semicircle = 14√2/2 cm = 7√2 cm

Area of ΔABC =( ½)×14×14 = 98 cm2

Area of quadrant = (¼)×(22/7)×(14×14) = 154 cm2

Area of the semicircle = (½)×(22/7)×7√2×7√2 = 154 cm2

Area of the shaded region =Area of the semicircle + Area of ΔABC – Area of quadrant

= 154 +98-154 cm2 = 98cm2

16. Calculate the area of the designed region in Fig. 12.34 common between the two quadrants of circles of radius 8 cm each.

Solution:

AB = BC = CD = AD = 8 cm

Area of ΔABC = Area of ΔADC = (½)×8×8 = 32 cm2

Area of quadrant AECB = Area of quadrant AFCD = (¼)×22/7×82

= 352/7 cm2

Area of shaded region = (Area of quadrant AECB – Area of ΔABC) = (Area of quadrant AFCD – Area of ΔADC)

= (352/7 -32)+(352/7- 32) cm2

= 2×(352/7-32) cm2

= 256/7 cm2

12.Areas related to the circles

Exercise 12.1

exercise 12.2

Exercise 12.3

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