Calculate the shaded areas in the shapes in Fig. 14.33. All dimensions are in centimetres. (Use the value 22/7 for pi.)

Here are the calculations for the shaded areas in Fig. 14.33, using $\pi = \frac{22}{7}$. All dimensions are in centimeters.

a) Shape a: A square with a circular hole.

Step 1: Calculate the area of the square.
The side length of the square is 10 cm.

$$\text{Area of square} = (10 \text{ cm})^2 = 100 \text{ cm}^2$$

Step 2: Calculate the area of the circle.
The diameter of the circle is 7 cm, so the radius is $r = \frac{7}{2}$ cm.

$$\text{Area of circle} = \pi r^2 = \frac{22}{7} \times \left(\frac{7}{2} \text{ cm}\right)^2 = \frac{22}{7} \times \frac{49}{4} \text{ cm}^2 = \frac{11 \times 7}{2} \text{ cm}^2 = 38.5 \text{ cm}^2$$

Step 3: Calculate the shaded area.

$$\text{Shaded area} = \text{Area of square} - \text{Area of circle} = 100 \text{ cm}^2 - 38.5 \text{ cm}^2 = \boxed{61.5 \text{ cm}^2}$$

b) Shape b: An annulus (ring).

Step 1: Calculate the area of the outer circle.
The outer diameter is 14 cm, so the outer radius is $R = \frac{14}{2} = 7$ cm.

$$\text{Area of outer circle} = \pi R^2 = \frac{22}{7} \times (7 \text{ cm})^2 = \frac{22}{7} \times 49 \text{ cm}^2 = 22 \times 7 \text{ cm}^2 = 154 \text{ cm}^2$$

Step 2: Calculate the area of the inner circle.
The inner diameter is 7 cm, so the inner radius is $r = \frac{7}{2}$ cm.

$$\text{Area of inner circle} = \pi r^2 = \frac{22}{7} \times \left(\frac{7}{2} \text{ cm}\right)^2 = \frac{22}{7} \times \frac{49}{4} \text{ cm}^2 = \frac{11 \times 7}{2} \text{ cm}^2 = 38.5 \text{ cm}^2$$

Step 3: Calculate the shaded area.

$$\text{Shaded area} = \text{Area of outer circle} - \text{Area of inner circle} = 154 \text{ cm}^2 - 38.5 \text{ cm}^2 = \boxed{115.5 \text{ cm}^2}$$

c) Shape c: A quarter circle.

Step 1: Calculate the area of the quarter circle.
The radius of the quarter circle is $r = 14$ cm.

$$\text{Shaded area} = \frac{1}{4} \pi r^2 = \frac{1}{4} \times \frac{22}{7} \times (14 \text{ cm})^2 = \frac{1}{4} \times \frac{22}{7} \times 196 \text{ cm}^2 = \frac{1}{4} \times 22 \times 28 \text{ cm}^2 = 22 \times 7 \text{ cm}^2 = \boxed{154 \text{ cm}^2}$$

d) Shape d: Two semicircles joined.

Step 1: Calculate the area of the two semicircles.
Each semicircle has a diameter of 7 cm, so the radius is $r = \frac{7}{2}$ cm. Two semicircles form one full circle.

$$\text{Shaded area} = \pi r^2 = \frac{22}{7} \times \left(\frac{7}{2} \text{ cm}\right)^2 = \frac{22}{7} \times \frac{49}{4} \text{ cm}^2 = \frac{11 \times 7}{2} \text{ cm}^2 = \boxed{38.5 \text{ cm}^2}$$

e) Shape e: A rectangle with two semicircles removed.

Step 1: Calculate the area of the rectangle.
The width of the rectangle is 14 cm and the height is 20 cm.

$$\text{Area of rectangle} = 14 \text{ cm} \times 20 \text{ cm} = 280 \text{ cm}^2$$

Step 2: Calculate the area of the two semicircles.
Each semicircle has a diameter of 14 cm, so the radius is $r = \frac{14}{2} = 7$ cm. Two semicircles form one full circle.

$$\text{Area of two semicircles} = \pi r^2 = \frac{22}{7} \times (7 \text{ cm})^2 = \frac{22}{7} \times 49 \text{ cm}^2 = 22 \times 7 \text{ cm}^2 = 154 \text{ cm}^2$$

Step 3: Calculate the shaded area.

$$\text{Shaded area} = \text{Area of rectangle} - \text{Area of two semicircles} = 280 \text{ cm}^2 - 154 \text{ cm}^2 = \boxed{126 \text{ cm}^2}$$

f) Shape f: A rectangle with a semicircle removed.

Step 1: Calculate the area of the rectangle.
The width of the rectangle is 5 cm and the height is 9 cm.

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