**Exercise 21.1 **

** **

**Question :1. Find the circumference of the circle whose radius is**

**(i) 14cm**

**(ii) 10m**

**(iii) 4km**

**Solution 1:**

**Question :2. Find the circumference of a circle whose diameter is**

**(i) 7 cm**

**(ii) 4.2 cm**

**(iii) 11.2 km**

**Solution 2:**

**Question :3. Find the radius of a circle whose circumference is**

**(i) 52.8 cm**

**(ii) 42 cm**

**(iii) 6.6 km**

**Solution 3:**

**Question :4. Find the diameter of a circle whose circumference is**

**(i) 12.56 cm**

**(ii) 88 m**

**(iii) 11.0 km**

**Solution 4:**

**Question :5. The ratio of the radii of two circles is 3: 2. What is the ratio of their circumferences?**

**Solution 5:**

_{1}= 2 π × 3r = 6 π r … (i)

_{2}= 2 × 2 π r = 4 π r … (ii)

_{1}/C

_{2}= ((6πr))/((4πr)) = 6/4 = 3/2

_{1}: C

_{2}

**Question :6. A wire in the form of a rectangle 18.7 cm long and 14.3 cm wide is reshaped and bent into the form of a circle. Find the radius of the circle so formed.**

**Solution 6:**

**Question :7. A piece of wire is bent in the shape of an equilateral triangle of each side 6.6 cm. It is re-bent to form a circular ring. What is the diameter of the ring?**

**Solution 7:**

**Question :8. The diameter of a wheel of a car is 63 cm. Find the distance travelled by the car during the period, the wheel makes 1000 revolutions.**

**Solution 8:**

**Question :9. The diameter of a wheel of a car is 98 cm. How many revolutions will it make to travel 6160 meters.**

**Solution 9:**

**Question :10. The moon is about 384400 km from the earth and its path around the earth is nearly circular. Find the circumference of the path described by the moon in lunar month.**

**Solution 10:**

**Question :11. How long will John take to make a round of a circular field of radius 21 m cycling at the speed of 8 km/hr.?**

**Solution 11:**

**Question :12. The hour and minute hands of a clock are 4 cm and 6 cm long respectively. Find the sum of the distances travelled by their tips in 2 days.**

**Solution 12:**

**Question :13. A rhombus has the same perimeter as the circumference of the circle. If the side of the rhombus is 2.2m, find the radius of the circle.**

**Solution 13:**

**Question :14. A wire is looped in the form of a circle of radius 28 cm. It is re-bent into a square form. Determine the length of the side of the square.**

**Solution 14:**

**Question :15. A bicycle wheel makes 5000 revolutions in moving 11 km. Find the diameter of the wheel.**

**Solution 15:**

**Question :16. A boy is cycling such that the wheels of the cycle are making 140 revolutions per minute. If the diameter of the wheel is 60 cm, calculate the speed per hour with which the boy is cycling.**

**Solution 16:**

**Question :17. The diameter of the driving wheel of a bus is 140 cm. How many revolutions per minute must the wheel make in order to keep a speed of 66 km per hour?**

**Solution 17:**

**Question :18. A water sprinkler in a lawn sprays water as far as 7 m in all directions. Find the length of the outer edge of wet grass.**

**Solution 18:**

**Question :19. A well of diameter 150 cm has a stone parapet around it. If the length of the outer edge of the parapet is 660 cm. then find the width of the parapet.**

**Solution 19:**

**Question :20. An ox in a kolhu (an oil processing apparatus) is tethered to a rope 3 m long. How much distance does it cover in 14 rounds?**

**Solution 20:**

**Exercise 21.2**

**Question :1. Find the area of a circle whose radius is**

**(i) 7 cm**

**(ii) 2.1 m**

**(iii) 7 km**

**Solution 1:**

^{2}

^{2}

^{2}

^{2}

^{2}

^{2}

^{2}

^{2}

^{2}

**Question :2. Find the area of a circle whose diameter is**

**(i) 8.4 cm**

**(ii) 5.6 m**

**(iii) 7 km**

**Solution 2:**

^{2}

^{2}

^{2}

^{2}

^{2}

^{2}

^{2}

^{2}

^{2}

**Question :3. The area of a circle is 154 cm2. Find the radius of the circle.**

**Solution 3:**

^{2}

^{2}

^{2}

^{2}= ((154 × 7))/22

^{2}= 49

**Question :4. Find the radius of a circle, if its area is**

**(i) 4 π cm**

^{2}**(ii) 55.44 m**

^{2}**(iii) 1.54 km**

^{2}**Solution 4:**

^{2}

^{2}

^{2}

^{2}

^{2}= 4

^{2}

^{2}

^{2}

^{2}= ((55.44 × 7))/22

^{2}= 17.64 m

^{2}

^{2}

^{2}

^{2}

^{2}= ((1.54 × 7))/22

^{2}= 0.49 km

**Question :5. The circumference of a circle is 3.14 m, find its area.**

**Solution 5:**

^{2}

^{2}

^{2}

**Question :6. If the area of a circle is 50.24 m**

^{2}, find its circumference.**Solution 6:**

^{2}

^{2}

^{2}

^{2}= ((50.24 × 7))/22

^{2}= 15.985

**Question :7. A horse is tied to a pole with 28 m long string. Find the area where the horse can graze. (Take π = 22/7).**

**Solution 7:**

^{2}

^{2}

^{2 }

**Question :8. A steel wire when bent in the form of a square encloses an area of 121 cm**

^{2}. If the same wire is bent in the form of a circle, find the area of the circle.**Solution 8:**

^{2}

^{2}= 121

^{2}

^{2}

^{2}

**Question :9. A road which is 7 m wide surrounds a circular park whose circumference is 352 m. Find the area of road.**

**Solution 9:**

^{2}– π × (56)

^{2}

^{2}

**Question :10. Prove that the area of a circular path of uniform width h surrounding a circular region of radius r is πh (2r + h).**

**Solution 10:**

^{2}– πr

^{2}

^{2}+ π h

^{2}+ 2 π r h – π r

^{2}

**Question :11. The perimeter of a circle is 4πr cm. What is the area of the circle?**

**Solution 11:**

^{2}

^{2}

^{2}cm

^{2}

**Question :12. A wire of 5024 m length is in the form of a square. It is cut and made a circle. Find the ratio of the area of the square to that of the circle.**

**Solution 12:**

^{2}: πr

^{2}

^{2}/ πr

^{2}

**Question :13. The radius of a circle is 14 cm. Find the radius of the circle whose area is double of the area of the circle.**** **

**Solution 13:**

Let be A_{1} is the area of the circle whose radius is 14 cm.

Formula of the area of the circle = πr^{2}

Thus, A_{1} = π (14)^{2}

Let A_{2} and r_{2} be the area and radius of the second circle respectively whose area is double the area of circle A_{1}.

A_{2} = 2 A_{1}

⇒ π (r_{2})^{2} = 2 × π (14)^{2}

⇒ (r_{2})^{2} = 2 × (14)^{2}

⇒ r_{2} = 14√2 cm

So, the radius of the circle A_{2} is 14√2 cm.

**Question :14. The radius of one circular field is 20 m and that of another is 48 m. find the radius of the third circular field whose area is equal to the sum of the areas of two fields.**** **

**Solution 14:**

In the given information, Let A_{1 }= the area of the circle whose radius is 20 m

A_{2 }= the area of the circle whose radius is 48 m

Now we must determine the radius of the third circle whose area equals the number of the areas of the two fields.

So,

A_{3 }= A_{1} + A_{2}

⇒ π r^{2} = π (20)^{2} + π (48)^{2}

⇒ π r^{2} = π [(20)^{2} + (48)^{2}]

⇒ r^{2} = 400 + 2304

⇒ r = 52 m

Thus, radius = 52 m

** **

**Question :15. The radius of one circular field is 5 m and that of the other is 13 m. Find the radius of the circular field whose area is the difference of the areas of first and second field.**** **

**Solution 15:**

In the given information, let A_{1 }= the area of the circular field whose radius is 5 m

A_{2 }= the area of the circular field whose radius is 13 m

Now we must calculate the area of a circular field whose area equals the sum of the first and second fields' regions.

A_{3 }= A_{2} – A_{1}

⇒ π r^{2} = π (13)^{2} – π (5)^{2}

⇒ π r^{2} = π [(13)^{2} – (5)^{2}]

⇒ r^{2} = 169 – 25

⇒ r^{2} = 144

⇒ r = 12 m

So, the radius of the circular field is 12 m.

** **

**Question :16. Two circles are drawn inside a big circle with diameters 2/3rd and 1/3rd of the diameter of the big circle as shown in Fig. 18. Find the area of the shaded portion, if the length of the diameter of the circle is 18 cm.**

**Solution 16:**

According to the question, diameter of the big circle = 18 cm

Radius of the big circle = 9 cm

Area of the big circle, A = π r^{2} = π (9)^{2} = 81π cm^{2}

Let d_{1} = 2/3× 18 = 12 cm

r_{1} = 6 cm

Area of the circle, A_{1} = π r^{2} = π (6)^{2} = 36π cm^{2}

d_{2} = 1/3 × 18 = 6 cm

r_{2} = 3 cm

Area of the circle, A_{2} = π r^{2} = π (3)^{2} = 9π cm^{2}

Area of the shaded portion = A – (A_{1} + A_{2})

Area of the shaded portion = 81π – (36π + 9π) = 36π cm^{2}

** **

**Question :17. In Fig. 19, the radius of quarter circular plot taken is 2 m and radius of the flower bed is 2 m. Find the area of the remaining field.**

**Solution 17:**

Radius of flower bed = 2 m

Area of flower bed = π r^{2} = π (2)^{2} = 4π

Radius of the quarter circular plot = 2 m

Area of the quarter circular plot = (π r^{2})/4

Area of 4 quarter circular plots = 4 × (π r^{2})/4

= π r^{2 }

= π (2)^{2}

= 4π

Formula of the area of the rectangular region = Length x Breadth

Area of the rectangular region = 8 x 6 = 48 m^{2}

Area of the remaining field = Area of the rectangular region – (Area of 4 quarter circular plots + Area of the flower bed)

Area of the remaining field = 48 – (4π + 4π)

= 48 – 25.12

= 22.88 m^{2}

**Question :18. Four equal circles, each of radius 5 cm, touch each other as shown in Fig. 20. Find the area included between them. (Take π = 3.14).**

**Solution 18:**

According to the diagram, side of the square = 10 cm

Formula of area of the square = side x side

= 10 x 10 = 100 cm^{2}

Radius of the quarter circle = 5 cm

Area of the quarter circle = (π r^{2})/4

Area of 4 quarter circle = 4 × (π r^{2})/4

= π r^{2 }

= 3.14 × (5)^{2}

= 78.5 cm^{2}

Area included in the quarter circle = Area of the square – Area of the four quarter circles

Area included in them = (100 – 78.5) cm^{2 }

= 21.5 cm^{2}

** **

**Question :19. The area of circle is 100 times the area of another circle. What is the ratio of their circumferences?**** **

**Solution 19:**

Let be A_{1} and A_{2} the area of the circles and their circumference be c_{1} and c_{2 }respectively.

According to the question, A_{1} = 100 A_{2}

⇒ π (r_{1})^{2} = 100 × π (r_{2})^{2}

⇒ r_{1} = 10 r_{2}

⇒ r_{1}/r_{2} = 10/1 -(i)

After finding the ratios of the circumference;

C_{1}: C_{2} = 2πr_{1}: 2πr_{2}

C_{1}/C_{2} = (2πr_{1})/ (2πr_{2})

C_{1}/C_{2} = r_{1}/r_{2}

Putting the value of r_{1}/r_{2 }from equation (i)

C_{1}/C_{2} = 10/1

C_{1}: C_{2} = 10/1

So, the ratio of their circumferences is 10: 1.