Full solution

Q. Find the Area of the figure below, composed of a square and four semicircles. Rounded to the tenths place

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Answer

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Find Square Area: First, let's find the area of the square. If the side of the square is $s$ , then the area of the square is $s^2$ .
Calculate Semicircle Area: Now, we need to find the area of one semicircle. The area of a full circle is $\pi r^2$ , so the area of a semicircle is $(\pi r^2)/2$ .
Total Semicircles Area: Since there are $4$ semicircles, we multiply the area of one semicircle by $4$ to get the total area of the semicircles. So, the total area of the semicircles is $4 \times (\pi r^2)/2$ .
Find Total Area: To find the total area of the figure, we add the area of the square and the total area of the semicircles. So, the total area is $s^2 + 4 \times (\pi r^2)/2$ .
Substitute Radius: Assuming the diameter of the semicircles is equal to the side of the square, the radius of the semicircles is $s/2$ . We substitute $r$ with $s/2$ in the area formula for the semicircles.
Calculate Square Area: Now we calculate the area using the values. Let's say the side of the square is $10\,\text{cm}$ (just as an example). The area of the square is $10^2 = 100\,\text{cm}^2$ .
Calculate Semicircle Area: The area of one semicircle with radius $5\,\text{cm}$ (half of the side of the square) is $(\pi \times 5^2)/2 = (\pi \times 25)/2 = 12.5\pi\,\text{cm}^2$ .
Calculate Total Semicircles Area: The total area of the four semicircles is $4 \times 12.5\pi \, \text{cm}^2 = 50\pi \, \text{cm}^2$ .
Add Areas: Adding the area of the square and the semicircles gives us $100 \, \text{cm}^2 + 50\pi \, \text{cm}^2$ . Since $\pi$ is approximately $3.14$ , we calculate $50 \times 3.14 = 157 \, \text{cm}^2$ .
Calculate Total Area: The total area of the figure is $100\,\text{cm}^2 + 157\,\text{cm}^2 = 257\,\text{cm}^2$ . We round this to the tenths place, which gives us $257.0\,\text{cm}^2$ .