Contents

**GSEB Solutions for Class 10 mathematics – Areas Related to a Circle (English Medium)**

### Exercise-13.1

**Question 1:**

Find the circumference and the area of the circle whose radius is 8.4 cm.

**Solution :**

**Question 2:**

Find the circumference of the circle whose area is 38.5 m^{2}.

**Solution :**

**Question 3:**

The inner circumference of a circular race track is 44 m less than the outer circumference. If the outer circumference is 396 m, then find the width of the track.

**Solution :**

**Question 4:**

The radius of the wheel of a truck is 7 cm. It takes 250 revolution per minute. Find the speed of the truck in km/hr.

**Solution :**

### Exercise-13.2

**Question 1:**

An arc of a circle whose radius is 21 cm subtends an angle of measure 120 at the centre. Find the length of the arc and area of the sector.

**Solution :**

**Question 2:**

The radius of a circular ground is 63 m. There is 7 m wide road inside the ground as shown in figure 13.10. The blue coloured portion of the road, shown in figure 13.10 is to be repaired. If the rate of repair work of the road costs ₹ 25 per m^{2}, find the total cost of repair.

**Solution :**

**Question 3:**

A regular hexagon of side 10 cm is cut from a plane circular sheet of radius 10 cm as shown in the figure 13.11. Find the area of the remaining part of the sheet. (\(\sqrt{3}\) = 1.73) (π = 3.14).

**Solution :**

**Question 4:**

The length of a minute hand of a circular dial is 10 cm. Find the area of the sector formed by the present position and position five minute of the minute hand. (π = 3.14)

**Solution :**

**Question 5:**

The radius of a field in the form of a sector is 21 m. The cost of constructing a wall around the field is ₹ 1875 at the rate of ₹ 25 per meter. If it costs ₹ 10 per m^{2} to the field. what will be the cost of tilling the whole field ?

**Solution :**

**Question 6:**

The length of a side of a square field is 20 m. A cow is tied at the corner by means of a 6 m long rope. Find the area of the field which the cow can graze. Also find the increase in grazing area, if the length of the rope is increased by 2 m.(π = 3.14)

**Solution :**

**Question 7:**

A chord of a circle of radius 42 cm subtends an angle of measure 60 at the centre. Find the area of the minor segment of the circle. (\(\sqrt{3}\) = 1.73)

**Solution :**

**Question 8:**

A chord of a circle, of length 10 cm, subtends a right angle at the centre. Find the areas of the minor segment and the major segment formed by the chord. (π = 3.14)

**Solution :**

### Exercise-13.3

**Question 1:**

A rectangle whose length and breadth are 12 cm and 5 cm respectively is inscribed in a circle. Find the area of the blue coloured region, as shown in the figure 13.19.

**Solution :**

**Question 2:**

ABCD, square park, has each side of length 80 m. There is a flower bed at each corner in the form of a sector of radius 7 m, as shown in figure 13.20. Find the area of the remaining part of the park.

**Solution :**

**Question 3:**

What will be the cost of covering the in figure 13.21 with a silver foil if the ₹ 100 per m^{2} ? (π = 3.14)

**Solution :**

**Question 4:**

ABCD is a square plate of 1 m length. As shown in figure circles are drawn with their center at A, B, C, D respectively, each with radius equal to 42 cm. The blue coloured part at each corner, as shown in the figure 13.22 is cut. What is the area of the remaining portion of the plate ?

**Solution :**

**Question 5:**

\(\overline{OA}\) and \(\overline{OB}\) are two mutually perpendicular radii of a circle of radius 10.5 cm. D ∈ \(\overline{OB}\) and OD = 6 cm. Find the area of blue coloured region shown in figure 13.23.

**Solution :**

### Exercise-13

**Question 1:**

The area of a circular park is 616 m^{2}. There is a 3.5 m wide track around the park running parallel to the boundary. Calculate the cost of fencing on the outer circle at the rate of ₹ 5 per meter.

**Solution :**

**Question 2:**

A man is cycling in such a way that the wheels of the cycle are making 140 revolutions per minute. If the diameter of the wheel is 60 cm, then how much distance will he cover in 2 hours ?

**Solution :**

**Question 3:**

If a chord \(\overline{AB}\) of ⨀(O, 20) subtends right angle at O, find the area of the minor segrnent.

**Solution :**

**Question 4:**

There are two arcs \(\overset\frown{APB}\) of ⨀(O, OA) and \(\overset\frown{AQB}\) of (M, MA) as shown in figure 13.24. Find the area enclosed by two arcs.

**Solution :**

**Question 5:**

The length of a side of a square garden ABCD is 70 m. A minor segment of ⨀(O, OA) is drawn on each of two opposite sides for developing lawn, as shown in figure 13.24. Find the area of the lawn.

**Solution :**

**Question 6:**

ABCD is a square of side 20 cm. Find the area of blue coloured region formed by the semi-circles drawn on each side as shown in figure 13.26. (π = 3.14)

**Solution :**

**Question 7:**

On a circular table top of radius 30 cm a design is formed leaving an equilateral triangle inscribed in a circle. Find the area of the design. (π = 3.14)

**Solution :**

**Question 8:**

Figure 13.27 shows a kite formed by a square PQRS and an iscosceles right triangle ARB whose congruent sides are 5 cm long. \(\overset\frown{APB}\) is an arc of a ⨀ (R, 42). Find the area of the blue coloured region.

**Solution :**

**Question 9:**

In figure 13.28, ABCD is a square with sides having length 8 cm. Find the area of the blue coloured region. (π = 3.14)

**Solution :**

**Question 10:**

A circle is inscribed in ∆PQR where m∠Q = 90, PQ = 8 cm and QR = 6 cm. Find the area of the blue coloured region shown in figure 13.29. (π = 3.14)

**Solution :**

**Question 11:**

Select a proper option (a), (b), (c) or (d) from given options :

**Question 11(1):**

If an arc of a circle subtends an angle of measure θ at the centre, then the area of the minor sector is ……..

**Solution :**

**Question 11(2):**

The area of a sector is given by the formula ……

(r is the radius and l is the length of an arc.)

**Solution :**

**Question 11(3):**

\(\overline{OA}\) and \(\overline{OB}\) are the two mutually perpendicular radii of a circle having radius 9 cm. The area of the minor sector corresponding to ∠AOB is ………. cm^{2}. (π = 3.14)

**Solution :**

**Question 11(4):**

A sector subtends an angle of measure 120 at the centre of a circle having radius of 21 cm. The area of the sector is ……. cm^{2}

**Solution :**

**Question 11(5):**

If the area and the circumference of a circle are numerically equal, then r ………..

**Solution :**

**Question 11(6):**

The length of an arc subtending an angle of measure 60 at the centre of a circle whose area is 616 is ………….

**Solution :**

**Question 11(7):**

The area of a minor sector of ⨀(O, 15) is 150. The length of the corresponding arc is …… . (π = 3.14)

**Solution :**

**Question 11(8):**

If the radius of a circle is increased by 10 %, then corresponding increase in the area of the circle is ………. (π = 3.14)

**Solution :**

**Question 11(9):**

If the ratio of the area of two circles is 1 : 4, then the ratio of their circumference …………

**Solution :**

**Question 11(10):**

The area of the largest triangle inscribed in a semi-circle of radius 8 is …………

**Solution :**

**Question 11(11):**

If the circumference of a circle is 44 then the length of a side of a square inscribed in the circle is ……..

**Solution :**

**Question 11(12):**

The length of minute hand of a clock is 14 cm. If the minute hand moves from 2 to 11 on the circular dial, then area covered by it is ……… cm^{2.}

**Solution :**

**Question 11(13):**

The length of minute hand of a clock is 15 cm. If the minute hand moves for 20 minutes on a circular dial of a clock, the area covered by it is …….. cm^{2}. (π = 3.14)

**Solution :**