How do you calculate the area of a compound shape with a semicircle and a square?

To calculate the area of a compound shape with a semicircle and a square, find the areas separately and add them.

First, identify the dimensions of the square and the semicircle. Let's say the side length of the square is \( s \). The diameter of the semicircle will be equal to the side length of the square, so the radius \( r \) of the semicircle is \( \frac{s}{2} \).

Next, calculate the area of the square. The formula for the area of a square is:
\[ \text{Area of the square} = s^2 \]

Then, calculate the area of the semicircle. The formula for the area of a full circle is \( \pi r^2 \). Since we only need the area of a semicircle, we take half of that:
\[ \text{Area of the semicircle} = \frac{1}{2} \pi r^2 \]
Substitute \( r = \frac{s}{2} \) into the formula:
\[ \text{Area of the semicircle} = \frac{1}{2} \pi \left( \frac{s}{2} \right)^2 = \frac{1}{2} \pi \left( \frac{s^2}{4} \right) = \frac{\pi s^2}{8} \]

Finally, add the areas of the square and the semicircle to get the total area of the compound shape:
\[ \text{Total area} = s^2 + \frac{\pi s^2}{8} \]

This method ensures you correctly account for both parts of the compound shape.