Chapter 12 Areas Related to circles

Case Study 1: Area of a Circle

Case Description:
A circular garden has a radius of 7 meters. The gardener wants to know the area of the garden to plan the amount of grass seed needed for planting. The area of a circle can be calculated using the formula $A = \pi r^2$, where $r$ is the radius of the circle. The gardener wants to ensure that he purchases enough grass seed to cover the entire area.

MCQs:

1. What is the radius of the circular garden in meters?

   • A) 14
   • B) 7
   • C) 21
   • D) 28

2. Using $\pi \approx 3.14$, what is the area of the circular garden?

   • A) 154 m²
   • B) 49 m²
   • C) 100 m²
   • D) 200 m²

3. If grass seed costs ₹5 per square meter, how much will the gardener spend on grass seed for the entire garden?

   • A) ₹770
   • B) ₹850
   • C) ₹700
   • D) ₹750

4. What will be the effect of increasing the radius of the garden to 10 meters on the area?

   • A) The area will decrease.
   • B) The area will remain the same.
   • C) The area will increase.
   • D) The area will become negative.

---

Case Study 2: Area of a Sector

Case Description:
A pizza is cut into 8 equal slices. Each slice forms a sector of a circle with a radius of 12 cm. To determine the area of each slice, the chef uses the formula for the area of a sector: $A = \frac{\theta}{360} \times \pi r^2$, where $\theta$ is the angle of the sector in degrees and $r$ is the radius. Since the pizza is divided into 8 slices, the angle of each sector will be $45^\circ$.

MCQs:

1. What is the angle of each sector in degrees?

   • A) 60
   • B) 30
   • C) 45
   • D) 90

2. What is the radius of the pizza in centimeters?

   • A) 8
   • B) 10
   • C) 12
   • D) 14

3. Using $\pi \approx 3.14$, what is the area of one slice of pizza?

   • A) 28.26 cm²
   • B) 18.84 cm²
   • C) 16.56 cm²
   • D) 45.84 cm²

4. If the pizza is sold at ₹15 per slice, how much will the total revenue be from selling all 8 slices?

   • A) ₹90
   • B) ₹120
   • C) ₹105
   • D) ₹75

---

Case Study 3: Area of a Ring

Case Description:
A circular swimming pool has a radius of 10 meters, and it is surrounded by a walkway that is 2 meters wide. The area of the walkway (ring) can be calculated by finding the area of the larger circle (pool plus walkway) and subtracting the area of the smaller circle (just the pool). The radius of the larger circle is $10 + 2 = 12$ meters.

MCQs:

1. What is the radius of the larger circle (pool plus walkway) in meters?

   • A) 10
   • B) 12
   • C) 14
   • D) 16

2. What is the area of the larger circle using $\pi \approx 3.14$?

   • A) 376.8 m²
   • B) 452.16 m²
   • C) 314 m²
   • D) 314.16 m²

3. What is the area of the smaller circle (the pool) using $\pi \approx 3.14$?

   • A) 314 m²
   • B) 328 m²
   • C) 250 m²
   • D) 300 m²

4. What is the area of the walkway (ring)?

   • A) 62.56 m²
   • B) 100.16 m²
   • C) 30.56 m²
   • D) 50.24 m²

---

Case Study 4: Circumference of a Circle

Case Description:
A circular track has a diameter of 100 meters. The sports coach wants to find the distance around the track, which is given by the circumference formula $C = \pi d$, where $d$ is the diameter of the circle. This information will help the coach plan the training sessions for the athletes.

MCQs:

1. What is the diameter of the circular track in meters?

   • A) 50
   • B) 75
   • C) 100
   • D) 125

2. Using $\pi \approx 3.14$, what is the circumference of the circular track?

   • A) 314 m
   • B) 150 m
   • C) 250 m
   • D) 400 m

3. If an athlete runs 5 laps around the track, what is the total distance covered?

   • A) 1250 m
   • B) 1570 m
   • C) 300 m
   • D) 750 m

4. How would the circumference change if the diameter were doubled?

   • A) It would be halved.
   • B) It would remain the same.
   • C) It would double.
   • D) It would increase by 25%.

---

Case Study 5: Composite Areas

Case Description:
A circular flower bed with a radius of 3 meters is surrounded by a rectangular pathway that measures 1 meter in width. The total area that needs to be covered with soil includes both the circular flower bed and the rectangular pathway. The area of the circular flower bed can be calculated using $A = \pi r^2$, and the area of the rectangular pathway can be calculated by subtracting the area of the flower bed from the area of the larger rectangle.

MCQs:

1. What is the radius of the circular flower bed in meters?

   • A) 2
   • B) 3
   • C) 4
   • D) 5

2. What is the area of the circular flower bed using $\pi \approx 3.14$?

   • A) 28.26 m²
   • B) 37.68 m²
   • C) 50.24 m²
   • D) 70.56 m²

3. If the width of the pathway is 1 meter, what is the radius of the larger circle that includes the pathway?

   • A) 3
   • B) 4
   • C) 5
   • D) 6

4. What is the total area of the flower bed and the pathway?

   • A) 37.68 m²
   • B) 50.24 m²
   • C) 62.56 m²
   • D) 78.54 m²