Circumference & Area of a Circle — Multiple Choice Questions

Choose the correct option:

If the ratio of the areas of two circles is 25 : 4, then the ratio of their diameters is :

1. 5 : 2

2. 25 : 4

3. 625 : 16

4. 16 : 625

Answer

Area of circle = πr².

Given,

Ratio of areas of two circles = 25 : 4

πR₁² : πR₂² = 25 : 4

R₁² : R₂² = 25 : 4

Taking square root on both terms

$\sqrt{R_1^2} : \sqrt{R_2^2} = \sqrt{25} : \sqrt{4}$

R₁ : R₂ = 5 : 2

Let R₁ = 5a and R₂ = 2a.

Diameter = 2 × Radius

D₁ = 2 × R₁ = 10a

D₂ = 2 × R₂ = 4a

D₁ : D₂ = 10a : 4a = 5 : 2.

Hence, option 1 is the correct option.

If the length of a side of a square is same as length of the diameter of a circle, then the ratio of their areas is :

1. 1 : π

2. 2 : π

3. 4 : π

4. 8 : π

Answer

Given,

Diameter of circle = side of square = s

Area of square = s²

Radius = $\dfrac{\text{Diameter}}{2} = \dfrac{s}{2}$

Area of circle = πr²

= $\pi \Big(\dfrac{s}{2}\Big)^2$

= $\dfrac{πs^2}{4}$

Area of square : Area of circle

= s² : $\dfrac{πs^2}{4}$

= $\dfrac{s^2}{\dfrac{πs^2}{4}}$

= $\dfrac{4}{π}$

= 4 : π.

Hence, option 3 is the correct option.

The ratio of the numerical values of the circumference and area of a semicircle of radius 5 units is :

1. 4 : 5

2. 2 : 5

3. 12 : 25

4. 6 : 25

Answer

Circumference of semicircle = πr = 5π.

$\text{Area of semicircle} = \dfrac{1}{2}πr^2 \\[1em] = \dfrac{1}{2} \times π \times 5^2 \\[1em] = \dfrac{25π}{2}$

Required ratio = $\dfrac{5π}{\dfrac{25π}{2}}$

= $\dfrac{10π}{25π}$

= $\dfrac{2}{5}$

= 2 : 5.

Hence, option 2 is the correct option.

If the perimeters of a circle and a square are same, then the ratio of their areas is :

1. 2 : π

2. 4 : π

3. 16 : π

4. π : 4

Answer

Given,

Circumference of circle = 2πr

Perimeter of square = 4a

Given,

Perimeter of square = Circumference of circle

⇒ 4a = 2πr

⇒ 2a = πr

⇒ a = $\dfrac{πr}{2}$

Area of circle = πr²

Area of square = a²

= $\Big(\dfrac{πr}{2}\Big)^2$

= $\dfrac{π^2r^2}{4}$

⇒ Area of circle : Area of square

= πr² : $\dfrac{π^2r^2}{4}$

= $\dfrac{πr^2}{\dfrac{π^2r^2}{4}}$

= $\dfrac{4πr^2}{π^2r^2}$

= $\dfrac{4}{π}$

= 4 : π.

Hence, option 2 is the correct option.

The area of the circumscribed circle of a square of each side p units is :

1. 2πp² sq units

2. $\dfrac{πp^2}{4}$ sq units

3. $\dfrac{πp^2}{2}$ sq units

4. πp² sq units

Answer

ABCD is a square with diagonal 'd' units and side 'p' units.

From figure,

Diameter of the circle = diagonal of the square.

Side of square = p units.

$\text{Diagonal of square (d)} = \sqrt{p^2 + p^2} \\[1em] = \sqrt{2p^2} \\[1em] = p\sqrt{2}$

Radius of circle = $\dfrac{d}{2} = \dfrac{p\sqrt{2}}{2}$ units.

Calculating,

$\text{Area of circle} = πr^2 \\[1em] = π\Big(\dfrac{p\sqrt{2}}{2}\Big)^2 \\[1em] = π \times \dfrac{p^2 × 2}{4} \\[1em] = \dfrac{πp^2}{2} \text{ sq units}.$

Hence, option 3 is the correct option.

The radius of a circle whose area is equal to the sum of the areas of two circles of radii 7 cm and 24 cm respectively is :

1. 31 cm

2. 28 cm

3. 27 cm

4. 25 cm

Answer

Let A₁ and A₂ be the areas of two circles.

Area = πr²

Areas of the two circles:

⇒ A₁ = π.(7)² = 49π.

⇒ A₂ = π.(24)² = 576π

∴ Sum of the areas of the two circles = 49π + 576π = 625π.

Let the radius of new circle be 'R' cm.

⇒ πR² = 625π

⇒ R² = 625

⇒ R = $\sqrt{625}$ = 25 cm.

Hence, option 4 is the correct option.

The radius of the circle whose area is equal to the sum of areas of two circles of radii 9 cm and 12 cm respectively is :

1. 11 cm

2. 12 cm

3. 14 cm

4. 15 cm

Answer

Let A₁ and A₂ be the areas of two circles.

Area = πr²

Areas of the two circles:

⇒ A₁ = π.(9)² = 81π.

⇒ A₂ = π.(12)² = 144π

∴ Sum of the areas of the two circles = 81π + 144π = 225π.

Let the radius of new circle be R cm.

⇒ πR² = 225π

⇒ R² = 225

⇒ R = $\sqrt{225}$ = 15 cm.

Hence, option 4 is the correct option.

The area of a circular garden is 55.44 m². How long wire is needed for fencing the garden ?

1. 21.4 m

2. 22.4 m

3. 24.6 m

4. 26.4 m

Answer

Given,

Area of circle = 55.44 m².

By formula,

$\text{Area of circle} = πr^2 \\[1em] \Rightarrow 55.44 = πr^2 \\[1em] \Rightarrow \dfrac{22}{7} \times r^2 = 55.44 \\[1em] \Rightarrow r^2 = \dfrac{55.44 × 7}{22} \\[1em] \Rightarrow r^2 = 17.64 \\[1em] \Rightarrow r = \sqrt{17.64} \\[1em] \Rightarrow r = 4.2 \text{ m}.$

Wire length = Circumference of the circle

= 2πr

= 2 × $\dfrac{22}{7}$ × 4.2

= 26.4 m.

Hence, option 4 is the correct option.

The sum of the lengths of a semicircular bow and its string is 360 cm. The length of the bow is :

1. 214 cm

2. 216 cm

3. 218 cm

4. 220 cm

Answer

Given,

The sum of the lengths of a semicircular bow and its string is 360 cm.

∴ Arc length + Diameter = 360

$\Rightarrow πr + 2r = 360 \\[1em] \Rightarrow \dfrac{22}{7} \times r + 2r = 360 \\[1em] \Rightarrow \dfrac{22r + 14r}{7} = 360 \\[1em] \Rightarrow \dfrac{36r}{7} = 360 \\[1em] \Rightarrow r = 70 \text{ cm}.$

Length of the bow = πr

= $\dfrac{22}{7}$ × 70

= 220 cm.

Hence, option 4 is the correct option.

Each side of a square formed by a wire is 14 cm. The area of the circle that can be formed by this wire is :

1. 144 cm²

2. 154 cm²

3. 164 cm²

4. 249.45 cm²

Answer

Given,

Each side of a square formed by a wire is 14 cm.

Total length of wire = Perimeter of square = 4 × 14 = 56 cm.

The circle formed with the wire will be having the circumference = 56 cm.

Let radius of circle be r cm.

2πr = 56

r = $\dfrac{56}{2π} = \dfrac{28}{π}$

Area of circle = πr²

$= \pi \times \left(\dfrac{28}{π}\right)^2 \\[1em] = \pi \times \dfrac{28}{π} \times \dfrac{28}{π} \\[1em] = \dfrac{28 \times 28}{π} \\[1em] = \dfrac{28 \times 28}{\dfrac{22}{7}} \\[1em] = \dfrac{28 \times 28 \times 7}{22} \\[1em] = 249.45 \text{ cm}^2$

Hence, option 4 is the correct option.

If the difference between the circumference and the diameter of a circle is 30 cm, the circumference of the circle is :

1. 44 cm

2. 45 cm

3. 46 cm

4. 48 cm

Answer

Given,

Circumference of circle - Diameter of circle = 30

$\Rightarrow 2πr - 2r = 30 \\[1em] \Rightarrow 2r(π - 1) = 30 \\[1em] \Rightarrow 2r\Big(\dfrac{22}{7} - 1\Big) = 30 \\[1em] \Rightarrow 2r(\dfrac{22 - 7}{7}) = 30 \\[1em] \Rightarrow 2r \times \dfrac{15}{7} = 30 \\[1em] \Rightarrow r = \dfrac{30 \times 7}{2 \times 15} \text{ cm}.$

Circumference of circle = 2πr

= 2 × $\dfrac{22}{7}$ × 7

= 2 × 22

= 44 cm.

Hence, option 1 is the correct option.

If the area of the inscribed circle of a square is 154 cm², then the area of a square is :

1. 190 cm²

2. 192 cm²

3. 196 cm²

4. 198 cm²

Answer

ABCD is a square with inscribed circle having radius 'r' and center O.

Given,

Area of circle = 154 cm².

We know that,

$\text{Area of circle} = πr^2 \\[1em] \Rightarrow πr^2 = 154 \\[1em] \Rightarrow \dfrac{22}{7} \times r^2 = 154 \\[1em] \Rightarrow r^2 = \dfrac{154 × 7}{22} \\[1em] \Rightarrow r^2 = 49 \\[1em] \Rightarrow r = \sqrt{49} = 7 \text{ cm}.$

For a circle inscribed in a square,

Diameter of circle = Side of square

Side = 2r = 2 × 7 = 14 cm.

Area of square = (side)²

= (14)²

= 196 cm².

Hence, option 3 is the correct option.

If the total cost of mowing a circular field at the rate of ₹1.20 per square metre is ₹4,620, then the cost of fencing the field at the rate of ₹4 per metre is :

1. ₹ 878

2. ₹ 880

3. ₹ 882

4. ₹ 884

Answer

Given,

Rate of mowing = ₹ 1.20 per square metre.

Total cost = ₹4,620

Area = $\dfrac{\text{Total cost}}{\text{Rate}}$

= $\dfrac{4620}{1.20}$ = 3850 m².

Let radius of circular field be r meters.

$\Rightarrow πr^2 = 3850 \\[1em] \Rightarrow \dfrac{22}{7} \times r^2 = 3850 \\[1em] \Rightarrow r^2 = \dfrac{3850 \times 7}{22} \\[1em] \Rightarrow r^2 = 1225 \\[1em] \Rightarrow r = \sqrt{1225} \\[1em] \Rightarrow r = 35 \text{ m}.$

Circumference of circle = 2πr

= 2 × $\dfrac{22}{7}$ × 35

= 220 m.

Cost of fencing = Circumference of field × Rate of fencing

= 220 × 4 = ₹ 880.

Hence, option 2 is the correct option.

There is a road of equal width all around a circular garden. The outer and inner circumferences of the road are 328 m and 200 m respectively. The area of the road will be :

1. 5376 m²

2. 5375 m²

3. 5374 m²

4. 5373 m²

Answer

Outer circumference (C₁) = 328 m

Inner circumference (C₂) = 200 m

Circumference = 2π.radius

Let outre radius be R meters and inner radius be r meters.

Calculating outer circumference,

$\Rightarrow \text{Outer circumference} = 2 \times \dfrac{22}{7} \times R \\[1em] \Rightarrow 328 = 2 \times \dfrac{22}{7} \times R \\[1em] \Rightarrow R = \dfrac{328 \times 7}{44} = \dfrac{574}{11} \text{ m}$

Calculating inner circumference,

$\Rightarrow \text{Inner circumference} = 2 \times \dfrac{22}{7} \times r \\[1em] \Rightarrow 200 = 2 \times \dfrac{22}{7} \times r \\[1em] \Rightarrow r = \dfrac{200 \times 7}{44} \\[1em] \Rightarrow r = \dfrac{350}{11} \text{ m}.$

Area = π(R² - r²)

= π(R + r)(R - r)

$= \dfrac{22}{7} \times \Big(\dfrac{574}{11} + \dfrac{350}{11}\Big) \times \Big(\dfrac{574}{11} - \dfrac{350}{11}\Big) \\[1em] = \dfrac{22}{7} \times \dfrac{924}{11} \times \dfrac{224}{11} \\[1em] = 2 \times 84 \times 32 \\[1em] = 5376 \text{ m}^2.$

Hence, option 1 is the correct option.

The diameter of the front wheel and the rear wheel of the cycle are 70 cm and 168 cm respectively. In covering a certain distance, the front wheel makes 600 revolutions. The number of revolutions made by the rear wheel to cover the same distance is :

1. 248

2. 250

3. 252

4. 254

Answer

Circumference of circle = 2πr = πd

Front wheel circumference :

C₁ = π × 70

Rear wheel circumference :

C₂ = π × 168

Distance covered by front wheel in 600 revolutions = 600 × π × 70

Let rear wheel revolutions = x

Since both cover the same distance

∴ x × π × 168 = 600 × π × 70

x = $\dfrac{600 × 70}{168}$

x = $\dfrac{42000}{168}$ = 250.

Hence, option 2 is the correct option.