Chapter 11 · Areas Related to Circles

NCERT Class 10 Maths Exercise 11.1

All 14 questions of NCERT Class 10 Maths Exercise 11.1 (Areas Related to Circles), in the book's order, each with its answer and a step-by-step solution.

1. Q1
   Find the ​area of a sector of a ​circle ​with radius ​6 cm if angle of the sector is ​60°.
   Show answer

   Area of the sector $=6\pi$ cm${}^2=\frac{132}{7}\approx 18.86$ cm${}^2$.

   Step-by-step solution →
2. Q2
   Find the area of a ​quadrant of a ​circle whose ​circumference is $22$ ​cm.
   Show answer

   Area of the quadrant $=\frac{77}{8}$ cm${}^2=9.625$ cm${}^2$.

   Step-by-step solution →
3. Q3
   The ​length of the ​minute ​hand ​of a clock is ​14 cm. Find ​the area swept by ​the minute ​hand in 5 minutes.
   Show answer

   Area swept $=\frac{154}{3}$ cm${}^2\approx 51.33$ cm${}^2$.

   Step-by-step solution →
4. Q4
   A chord ​of ​a ​circle of radius $10$ cm subtends a right angle at the centre. Find the area ​of the corresponding:

   1. (i)minor ​segment, ​
   2. (ii)​major sector. (Use $\pi =3.14$)
   Show answer

   (i) Area of the minor segment $=28.5$ cm${}^2$. (ii) Area of the major sector $=235.5$ cm${}^2$.

   Step-by-step solution →
5. Q5
   In a circle of ​radius ​21 cm, ​an arc subtends an angle of 60° at the centre. Find:

   1. (i)the length of the arc
   2. (ii)area of the sector formed by the arc ​
   3. (iii)​area ​of the ​segment formed by the corresponding ​chord.
   Show answer

   (i) Arc $=22$ cm. (ii) Sector area $=231$ cm${}^2$. (iii) Segment area $=231-\frac{441\sqrt{3}}{4}\approx 40.05$ cm${}^2$.

   Step-by-step solution →
6. Q6
   A ​chord of a circle of radius 15 ​cm subtends an angle of $60^{\circ }$ at the centre. Find the areas of ​the corresponding minor and major segments of the ​circle. (Use $\pi =3.14$ and $\sqrt{3}=1.73$) ​

   1. (i)The area ​of ​the corresponding minor segment.
   2. (ii)The area of ​the corresponding major segment.
   Show answer

   (i) Area of the minor segment $\approx 20.44$ cm${}^2$. (ii) Area of the major segment $\approx 686.06$ cm${}^2$.

   Step-by-step solution →
7. Q7
   A ​chord of a ​circle ​of ​radius ​12 cm subtends an ​angle of $120^{\circ }$ at the centre. Find the area of the corresponding segment of the circle. (Use $\pi =3.14$ ​and $\sqrt{3}=1.73$)
   Show answer

   Area of the minor segment $\approx 88.44$ cm${}^2$.

   Step-by-step solution →
8. Q8
   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. 11.8). Find

   1. (i)the area of that part of the field in which ​the horse ​can graze.
   2. (ii)the increase in the grazing area ​if the rope were ​10 m long instead ​of 5 ​m. (Use $\pi =3.14$)

   Show answer

   (i) Grazing area $=19.625$ m${}^2$. (ii) Increase in grazing area $=58.875$ m${}^2$.

   Step-by-step solution →
9. Q9
   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. 11.9. ​Find:

   1. (i)the total length of the silver wire required.
   2. (ii)the area of each sector of the brooch.

   Show answer

   (i) Total silver wire $=285$ mm. (ii) Area of each sector $=\frac{385}{4}$ mm${}^2=96.25$ mm${}^2$.

   Step-by-step solution →
10. Q10
    An umbrella has ​8 ribs which are ​equally spaced ​(see Fig. 11.10). Assuming umbrella to be ​a flat circle of radius 45 cm, find the area ​between ​the two consecutive ​ribs of the umbrella.

    Show answer

    Area between two consecutive ribs $=\frac{22275}{28}$ cm${}^2\approx 795.54$ cm${}^2$.

    Step-by-step solution →
11. Q11
    A car ​has two wipers which do not overlap. Each wiper has a blade ​of length 25 ​cm sweeping ​through an ​angle of $115^{\circ }$. Find the ​total area cleaned at ​each sweep of the blades.
    Show answer

    Total area cleaned $\approx 1254.96$ cm${}^2$ $\left(=\frac{158125}{126}{\text{ cm}}^2\right)$.

    Step-by-step solution →
12. Q12
    To warn ​ships for ​underwater rocks, a ​lighthouse spreads a red coloured light ​over a sector of angle $80^{\circ }$ to a ​distance of $16.5$ km. Find the ​area of the sea over which the ships ​are warned. ​(Use $\pi =3.14$)
    Show answer

    Area of the sea warned $\approx 189.97$ km${}^2$.

    Step-by-step solution →
13. Q13
    A round table cover has six equal designs as ​shown in Fig. 11.11. ​If the radius ​of the cover is 28 ​cm, find the cost ​of making the ​designs at the rate of $\text{Rs }0.35$ ​per cm${}^2$. ​(Use $\sqrt{3}=1.7$)

    Show answer

    Total cost of making the six designs $=\text{Rs }162.68$.

    Step-by-step solution →
14. Q14
    Tick ​the correct answer ​in the following: ​Area ​of ​a ​sector of angle $p$ ​(in degrees) of a circle with radius R ​is

    1. (a)$\frac{p}{180}\times 2\pi R$
    2. (b)$\frac{p}{180}\times \pi R^2$
    3. (c)$\frac{p}{360}\times 2\pi R$
    4. (d)$\frac{p}{720}\times 2\pi R^2$
    Show answer

    (d) $\frac{p}{720}\times 2\pi R^2$

    Step-by-step solution →