Chapter 11: Areas Related to Circles

A group of friends ordered two circular pizzas of the same size for a party. Each pizza has a diameter of 42 cm. One pizza was cut into 6 equal sectors (slices) and the other into 4 equal sectors.

1. What is the area of one slice from the 6-slice pizza?
   (A) 231 cm$^2$

   (B) 462 cm$^2$

   (C) 154 cm$^2$

   (D) 308 cm$^2$
   Show Solution
   Solution: (A) 231 cm$^2$
   Step 1: Radius $r = 42/2 = 21$ cm. Angle $\theta = 360/6 = 60^\circ$.
   Step 2: Area $= \frac{60}{360} \times \frac{22}{7} \times 21 \times 21 = \frac{1}{6} \times 22 \times 3 \times 21 = 11 \times 21 = 231$ cm$^2$.
2. What is the difference in the perimeter of a slice from the 4-slice pizza and a slice from the 6-slice pizza?
   (A) 11 cm

   (B) 22 cm

   (C) 33 cm

   (D) 44 cm
   Show Solution
   Solution: (A) 11 cm
   Step 1: Perimeter of sector $= 2r + \text{arc length}$. $2r$ is same for both.
   Step 2: Difference in arc length $= \frac{90}{360} \times 2\pi r - \frac{60}{360} \times 2\pi r = (\frac{1}{4} - \frac{1}{6}) \times 2 \times \frac{22}{7} \times 21$.
   Step 3: $= \frac{1}{12} \times 44 \times 3 = \frac{132}{12} = 11$ cm.

To keep the windshield of a car clear during rain, two wipers are used. The wipers do not overlap. Each wiper has a blade of length 21 cm sweeping through an angle of $120^\circ$.

1. Find the total area cleaned at each sweep of the blades.
   (A) 462 cm$^2$

   (B) 924 cm$^2$

   (C) 1386 cm$^2$

   (D) 231 cm$^2$
   Show Solution
   Solution: (B) 924 cm$^2$
   Step 1: Area of one sector $= \frac{120}{360} \times \frac{22}{7} \times 21 \times 21 = \frac{1}{3} \times 22 \times 3 \times 21 = 462$ cm$^2$.
   Step 2: Total area $= 2 \times 462 = 924$ cm$^2$.
2. What is the length of the arc swept by one blade?
   (A) 22 cm

   (B) 44 cm

   (C) 66 cm

   (D) 88 cm
   Show Solution
   Solution: (B) 44 cm
   Step 1: Arc length $= \frac{\theta}{360} \times 2\pi r = \frac{120}{360} \times 2 \times \frac{22}{7} \times 21$.
   Step 2: $= \frac{1}{3} \times 44 \times 3 = 44$ cm.

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.

1. What is the total length of the silver wire required?
   (A) 175 mm

   (B) 285 mm

   (C) 110 mm

   (D) 300 mm
   Show Solution
   Solution: (B) 285 mm
   Step 1: Circumference $= \pi d = \frac{22}{7} \times 35 = 110$ mm.
   Step 2: Length of 5 diameters $= 5 \times 35 = 175$ mm.
   Step 3: Total length $= 110 + 175 = 285$ mm.
2. What is the area of each sector of the brooch?
   (A) 385/2 mm$^2$

   (B) 385/4 mm$^2$

   (C) 385 mm$^2$

   (D) 358/4 mm$^2$
   Show Solution
   Solution: (B) 385/4 mm$^2$
   Step 1: Radius $r = 35/2$ mm. Angle $\theta = 360/10 = 36^\circ$.
   Step 2: Area $= \frac{36}{360} \times \frac{22}{7} \times \frac{35}{2} \times \frac{35}{2} = \frac{1}{10} \times 11 \times 5 \times \frac{35}{2} = \frac{1}{2} \times 11 \times \frac{35}{2} = \frac{385}{4}$ mm$^2$.