concept academy maths class 10 700-726-76-88

CONCEPT ACADEMY MATHS CLASS 10 700-726-76-88

CONCEPT ACADEMY शशशशश शश शशशश शश शश.....

CLASS X

MATHEMATICS

Chapter 12: Areas related to Circles

OUR CONCEPTS


CONCEPT’S QUESTIONS

1. In the figure, find teh perimeter of the shaded region where. ADC, AEB and BFC are semi-circles on diameters AC, AB and

BC respectively.

2. In the figure, arcs and drawn by taking vertices A, B and C of an equilateral triangle of side 10 cm to intersect the sides BC,

CA and AB at their respective mid-points D, E and F. Find the area of teh shaded region. [use π = 3.14]

3. In figure OABC is a quadrant of a circle of radius 7 cm. If OD = 4 cm, find the area of the shaded region

4. A chord of a circle of radius 20 cm subtends an angle of 90° at the centre. Find the area of the corresponding major segment

of the circle.

5. Area of a sector of a circle of radius 36 cm is 54π cm2. Find the length of the corresponding arc of the sector.

6. In the figure, AB is a diameter of the circle, AC = 6 cm and BC = 8 cm. Find the area of the shaded region. [Use π = 3.14]

7. A calf is tied with a rope of length 6 m at the corner of a square grassy lawn of side 20 m. If the length of the rope is

increased by 5.5 m, find the increase in area of the grassy lawn in which the calf can graze.

8. In the figure, a circle is inscribed in an equilateral triangle ABC of side 12 cm. Find the radius of teh inscribed circle and the

area of the shaded region. [Use π = 3.14 and √3 = 1.73]

CONCEPT’S SAMPLE QUESTIONS

1. The area of a n equilateral triangle is 49 √3 cm2. Taking each angular point as centre, circles are drawn with radius equal to

half length of the side of the triangle. Find the area of triangle not included in the circles.

2. A square OABC is inscribed in a quadrant OABQ of a circle as shown in the figure. If OA = 14 cm, find the area of the

shaded region.

3. In the figure, ABDC is a quadrant of a circle of radius 14 cm and a semi-circle is drawn with diameter BC. Find the area of

the shaded region.

4. Four equal circle are described about the four corners of a square so that each touches two of the others, as shown in teh

figure. Find the area of the shaded region, each side of the square measuring 14 cm.


5. In the figure, MN and PQ are teh arcs of two concentric circles of radii 7 cm and 3.5 cm respectively and ∠MON = 30°.

Find the area of the shaded region.

6. A play ground is in the form of a rectangle havign semi-circles on the shorter sides as shown in the figure. Find its area

when the length of the rectangular portion is 38 m and the breadth is 10 m. [use π = 3.14]

7. A bicycle wheel makes revolutions per minute to maintain a speed of 8.91 km per hour.

Find the diameter of the wheel.

8. Find the area of the shaded region in the given figure, if PQ = 24 cm, PR = 7 cm and O is the centre of the circle.

9. In the figure, from a rectangular region ABCD with AB = 20 cm, a right raignale AED with AE = 9 cm and DE = 12 cm, is

cut off. On the other end, taking BC as diameter, a semi-circle is added on outside the region. Find the area of the shaded region

10. In the figure, two concentric circles with centre O, have radii 21 cm and 42 cm. If ∠AOB = 60°, find the area of the shaded

region.


NCERT SOLUTIONS (By Concept)

Exercise 12.1 Q.1. 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.

Sol. We have, r1 = 19 cm

r2 = 9 cm

∴ Circumference of circle-I = 2π r1 = 2π (19) cm

Circumference of circle-II = 2π r1 = 2π (19) 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 = 2π R

According to the condition,

2π R = 2π (28)

Thus, the radius of the new circle = 28 cm.

Q.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.

Sol. We have,

Radius of circle-I, r1 = 8 cm

∴ Area of circle-I = πr1 2 = π(8)2 cm2

Area of circle-II = r22 = π(6)2 cm2

Let the area of the circle-III be R

∴ Area of circle-III = πr2

Now, according to the condition,

πr12 + πr2

2 = πr2

i.e. π(8)2 + π(6)2 = πr2

⇒ π(82 + 62) = πr2

⇒ 82 + 62 = r2

⇒ 64 + 36 = r2

⇒ 100 = r2

⇒ 102 = r2 ⇒ R = 10

Thus, the radius of the new circle = 10 cm.

Q.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.


Q.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?

Sol. Diameter of a wheel = 80 cm

Distance travelled by the car in 1hr

= 66 km = 66 × 1000 × 100 cm

∴ Distance travelled in 10 minutes

Thus, the required number of revolutions = 4375.

Q.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 unites (D) 7 units

Sol. We have: [Numerical area of the circle] = [Numerical circumference of the circle]

⇒ πr2 = 2π r

⇒ πr2 – 2π r = 0

⇒ r2 – 2 r = 0

⇒ r(r – 2) = 0


⇒ r = 0 or r = 2

But r cannot be zero

∴ r = 2 units.

Thus, the option (A) 2 units is correct.

• Area of Sector and Segment of a Circle

The portion (or part) of the circular region enclosed by two radii and the corresponding are is called a sector of the circle.

The portion (or part) of the circular region enclosed between a chord and the corresponding arc is called a segment of the circle.

Exercise 12.2 Q.1. Find the area of a sector of a circle with radius 6 cm if angle of the sector is 60°.

Sol. Here, r = 6 cm

θ = 60°

Q.2. Find the area of a quadrant of a circle whose circumference is 22 cm.

Sol. Let radius of the circle = r

∴ 2πr = 22

∴ Area of the quadrant of the circle,

Q.3. The length of the minute hand of a clock is 14 cm. Find the area swept by the minute hand in 5 minutes.

Sol. [Length of minute hand] = [radius of the circle]

⇒ r = 14 cm

θ Angle swept by the minute hand in 60 minutes = 360°


Q.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 5m 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 10m long instead of 5m.

Sol. 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

(ii) When length of the rope is increased to 10 m,

∴ r = 10 m

⇒ Area of the circular region where θ = 90°.

Q.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.

Sol. Diameter of the circle = 35 mm

Length of 1 piece of wire used to make diameter to divide the circle into 10 equal sectors = 35 mm

∴ Length of 5 pieces = 5 × 35 = 175 mm

∴ Total length of the silver wire = 110 + 175 mm = 285 mm

(ii) Since the circle is divided into 10 equal sectors,


Q.10. An umbrella has 8 ribs which are equally spaced (see figure). Assuning umbrella to be a flat circle of radius 45 cm, find

the area between the two consecutive ribs of the umbrella.

Sol. Here, radius (r) = 45 cm

Since circle is divided in 8 equal parts,

∴ Sector angle corresponding to each part

concept academy maths class 10 700-726-76-88

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