Circumference & Area of a Circle — Competency Focused Questions

The number of rounds that a wheel of diameter $\dfrac{7}{11}$ m will make in going 4 km is :

1. 1600

2. 1800

3. 1900

4. 2000

Answer

Given,

Diameter of wheel (d) = $\dfrac{7}{11}$ m

Distance = 4 km = 4000 m.

Distance covered in 1 round = Circumference

= 2πr

= πd

= $\dfrac{22}{7} × \dfrac{7}{11}$

= 2 m.

Number of rounds = $\dfrac{\text{Total distance}}{\text{Distance covered in one round}}$

= $\dfrac{4000}{2}$

= 2000.

Hence, option 4 is the correct option.

Four circular cardboard pieces, each of radius 7 cm are placed in such a way that each piece touches two other pieces. The area of the space enclosed by the four pieces is :

1. 42 cm²

2. 40 cm²

3. 21 cm²

4. 18 cm²

Answer

Given,

Radius = 7 cm

From figure,

Distance between centres of two touching circles = 2r = 2 × 7 = 14 cm.

So side of the square : AB = BC = CD = DA = 14 cm.

Area of square = (side)²

= (14)² = 196 cm².

At each corner of the square there is a quarter circle of radius 7 cm.

Four quarter circles together make one full circle.

Area of one full circle = πr²

= $\dfrac{22}{7}$ × 7²

= 22 × 7 = 154 cm².

Area of enclosed space = Area of square - Area of four quarter circles

= 196 - 154

= 42 cm².

Hence, option 1 is the correct option.

In the figure, if the radius of each circle is 5 cm, then area of the shaded region is :

1. (400 - 50π) cm²

2. (400 + 100π) cm²

3. (400 - 100π) cm²

4. 231 cm²

Answer

There are 9 equal circles each with radius = 5 cm.The shaded region is the space between the four middle touching circles.

The centres of a the four middle circles form a square.

Diameter = 2r = 2 × 5 = 10 cm

Side of a square = 10 cm.

Area of square = (side)²

= 10² = 100 cm².

Inside the square there are 4 quarter circles, which together to form one full circle of radius 5 cm.

∴ Area of circle = πr²

= π × 5² = 25π.

Area of one shaded region = 100 - 25π.

∴ For 4 identical shaded region

Total area = 4(100 - 25π)

= 400 - 100π cm².

Hence, option 3 is the correct option.

If sum of the areas of two circles with radii r₁ and r₂ is equal to the area of circle of radius r, then :

1. r = r₁ + r₂

2. r > r₁ + r₂

3. r² < r₁² + r₂²

4. r² = r₁² + r₂²

Answer

Given,

Sum of the areas of two circles with radii r₁ and r₂ is equal to the area of circle of radius r.

π(r₁)² + π(r₂)² = πr²

r₁² + r₂² = r²

Hence, option 4 is the correct option.

Area of sector of central angle 200° of a circle is 770 cm². The length of the corresponding arc of this sector is :

1. $70\dfrac{1}{2}$ cm

2. $73\dfrac{1}{3}$ cm

3. 76 cm

4. $80\dfrac{1}{2}$ cm

Answer

Given,

Area of sector = 770 cm²

Central angle = 200°

We know that,

$\Rightarrow \text{Area of sector} = \dfrac{\theta}{360°} \times πr^2 \\[1em] \Rightarrow 770 = \dfrac{200°}{360°} \times \dfrac{22}{7} \times r^2 \\[1em] \Rightarrow 35 = \dfrac{5}{9} \times \dfrac{1}{7} \times r^2 \\[1em] \Rightarrow r^2 = 7 \times 9 \times 7 \\[1em] \Rightarrow r^2 = 441 \\[1em] \Rightarrow r = \sqrt{441} \\[1em] \Rightarrow r = 21 \text{ cm}.$

Calculating the arc length :

$\Rightarrow \text{Arc length} = \dfrac{\theta}{360°} \times 2πr \\[1em] = \dfrac{200°}{360°} \times 2 \times \dfrac{22}{7} \times 21 \\[1em] = \dfrac{5}{9} \times 132 \\[1em] = \dfrac{660}{9} \\[1em] = 73\dfrac{1}{3} \text{ cm}.$

Hence, option 2 is the correct option.

Check whether the following statement is true or false. Justify your answer.
If the length of an arc of a circle of radius r is equal to that of an arc of a circle of radius 2r, then the angle of the corresponding sector of the first circle is double the angle of the corresponding sector of the other circle.

Answer

We know that,

Arc length = $\dfrac{\theta}{360}$ × 2πr

Let L₁ and L₂ be the arc length of first circle and second circle and θ₁ and θ₂ be the central angle of the sector for the first circle and second circle respectively.

Arc length for first circle (L₁) = $\dfrac{\theta_1}{360°}$ × 2πr

Arc length for second circle (L₂) = $\dfrac{\theta_2}{360}°$ × 2π(2r)

L₂ = $\dfrac{\theta_2}{360°}$ × 4πr

According to question :

L₁ = L₂

$\dfrac{\theta_1}{360°} × 2πr = \dfrac{\theta_2}{360°}$ × 4πr

2θ₁ = 4θ₂

θ₁ = 2θ₂

∴ Angle of the corresponding sector of the first circle is double the angle of the corresponding sector of the other circle.

Hence, the statement is True.

In the figure, a circle is inscribed in a square of side 5 cm and another circle is circumscribing the square. Find the ratio of the area of the outer circle to that of the inner circle.

Answer

Given,

Side of square = 5 cm.

Let A₁ and A₂ be the areas of the inner and outer circle respectively.

Inner circle (inscribed in the square):

Diameter of the circle = side of the square

Diameter = 5 cm

So, radius = $\dfrac{5}{2}$ = 2.5 cm.

Area of inner circle (A₁) = πr²

= π(2.5)²

= 6.25π cm²

Outer circle (circumscribing the square)

Diameter of the outer circle = Diagonal of the square

Diameter of the outer circle = $a\sqrt{2} = 5\sqrt{2}$ cm.

Radius of the outer circle = $\dfrac{5\sqrt{2}}{2}$

Area of outer circle (A₂) = πr²

= $π\Big(\dfrac{5\sqrt{2}}{2}\Big)^2$

= π × $\dfrac{25 × 2}{4}$

= 12.5π cm².

Ratio of areas :

Outer area : Inner area

A₂ : A₁

12.5π : 6.25π

2 : 1.

Hence, ratio = 2 : 1.