$15$ The figure shows a large square of side $84 \mathrm{~cm} . \mathrm{A}, \mathrm{B}$ and $\mathrm{C}$ are mid-points of three sides of the square. The large square is made up of four smaller identical squares. Inside each of the smaller squares is a semicircle and a quarter circle. Find the total area of the shaded parts. (Take $\pi=\frac{22}{7}$ )

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15

The figure shows a large square of side

84 \mathrm{~cm} . \mathrm{A}, \mathrm{B}

and

\mathrm{C}

are mid-points of three sides of the square. The large square is made up of four smaller identical squares. Inside each of the smaller squares is a semicircle and a quarter circle. Find the total area of the shaded parts. (Take

\pi=\frac{22}{7}

)

Determine Side Length: The first step is to determine the side length of the smaller squares. Since $A$ , $B$ , and $C$ are mid-points, each side of the smaller squares is half the side of the large square. $\newline$ Calculation: Side of smaller square = $\frac{84 \, \text{cm}}{2} = 42 \, \text{cm}$
Calculate Area of Smaller Square: Next, we calculate the area of one smaller square. Calculation: Area of smaller square = $(\text{side length})^2 = (42 \, \text{cm})^2 = 1764 \, \text{cm}^2$
Total Area of Smaller Squares: Since there are four smaller squares, we calculate the total area of all four smaller squares. $\newline$ Calculation: Total area of four smaller squares = $4 \times 1764 \, \text{cm}^2 = 7056 \, \text{cm}^2$
Calculate Radius of Semicircle: Now, we need to find the area of the semicircle and quarter circle within one smaller square. The diameter of the semicircle is the same as the side of the smaller square, so the radius is half of that. $\newline$ Calculation: Radius of semicircle = $\frac{42\,\text{cm}}{2}$ = $21\,\text{cm}$
Area of Semicircle Calculation: We calculate the area of the semicircle using the radius. $\newline$ Calculation: Area of semicircle = $(1/2) \times \pi \times (\text{radius})^2 = (1/2) \times (22/7) \times (21 \, \text{cm})^2 = (1/2) \times (22/7) \times 441 \, \text{cm}^2 = 1386 \, \text{cm}^2$
Area of Quarter Circle Calculation: The quarter circle also has the same radius as the semicircle. We calculate its area. $\newline$ Calculation: Area of quarter circle = $(1/4) \times \pi \times (\text{radius})^2 = (1/4) \times (22/7) \times (21 \, \text{cm})^2 = (1/4) \times (22/7) \times 441 \, \text{cm}^2 = 693 \, \text{cm}^2$
Total Shaded Area in One Square: Each smaller square contains one semicircle and one quarter circle, so we add their areas to find the total area of the shaded parts in one smaller square. $\newline$ Calculation: Total shaded area in one smaller square = Area of semicircle + Area of quarter circle = $1386 \, \text{cm}^2 + 693 \, \text{cm}^2 = 2079 \, \text{cm}^2$
Total Shaded Area in Large Square: Since there are four smaller squares, we multiply the shaded area of one smaller square by four to find the total shaded area in the large square. $\newline$ Calculation: Total shaded area in large square = $4 \times 2079 \, \text{cm}^2 = 8316 \, \text{cm}^2$