Question

24. The figure is made up of a circle and 2 identical semicircles inside a bigger semicircle. O is the centre of the bigger semicircle. The radius of the bigger semicircle is 14 cm. Find the total area of the shaded parts. (Take n = 7 22.)

Answer 1

The total area of the shaded parts is 56π cm^2.

To find the total area of the shaded parts, we need to find the area of the bigger semicircle and subtract the area of the circle and the two identical semicircles.

Area of the bigger semicircle:

`A_b = 1/2 * \pi * r^2`

where r is the radius of the bigger semicircle, which is 14 cm.

`A_b = 1/2 * \pi * 14^2\\A_b = 154\pi cm^2`

Area of the circle:

`A_c = \pi * r^2`

where r is the radius of the circle, which is 7 cm.

`A_c = \pi * 7^2\\A_c = 49\pi cm^2`

Area of the two identical semicircles:

`A_s = 2 * 1/2 * \pi * r^2`

where r is the radius of the identical semicircles, which is 7 cm.

`A_s = \pi * 7^2\\A_s = 49\pi cm^2`

Total area of the shaded parts:

`A_(shaded) = A_b - A_c - A_s\\A_(shaded) = 154\pi cm^2 - 49\pi cm^2 - 49\pi cm^2\\A_(shaded) = 56\pi cm^2`

The total area of the shaded parts is 56π cm^2.

Answer 2

To calculate the shaded area, we need to define some parts of the figure and calculate their areas. Then, we must add/subtract areas to isolate the required parts.

We will focus on the figure proved as it is, i.e., with the custom labels and marks on it.

First, we need to calculate the area of the zone marked as G or C1.

Calculate area A1, which can be done by subtracting the area of the square of side 7 cm and the area of a quarter circle of radius 7 cm:

`\begin{gathered} A1=7^2-(\pi)/(4)\cdot7^2 \\ A1=49(1-(\pi)/(4)) \end{gathered}`

The sum of areas A and A1 (both are equal) is the unshaded area inside of the square. We call it A2:

`A2=98(1-(\pi)/(4))`

We are now ready to calculate area G, the area of the square minus A2:

`\begin{gathered} G=7^2-98(1-(\pi)/(4)) \\ G=49-98+(98\pi)/(4) \\ G=49((\pi)/(2)-1) \end{gathered}`

The final part is to calculate the greater shaded area. It can be found by subtracting half the area of the bigger semicircle and the area exterior to the smaller circles.

We can add up the areas of the smaller circles, but we need to subtract the common area that is repeated twice (that is, subtract 2G).

If we compose the areas of the smaller circles, we would have two circles of radii of 7 cm. The total area of the two circles is:

`\begin{gathered} C=2*\pi *7^2 \\ C=98\pi \end{gathered}`

The inner area of the smaller circles is:

`\begin{gathered} C_i=C-2G \\ Ci=98\pi-98((\pi)/(2)-1) \\ C_i=98((\pi)/(2)+1) \end{gathered}`

The external shaded area is:

`\begin{gathered} A_e=(\pi)/(2)*14^2-98((\pi)/(2)+1) \\ A_e=98((\pi)/(2)-1) \end{gathered}`

Finally, the total shaded area is:

`\begin{gathered} A=A_i+2G \\ A=98((\pi)/(2)-1)+98((\pi)/(2)-1) \\ A=98(\pi-2) \end{gathered}`

Substituting the value of pi = 22/7:

A ≈ 112 square cm