Question

a square is inscribed in a circle, as shown below. dimensions given are in centim square centimeter, what is the combined area of the four regions shaded pink?

Explanation:

Step1: Calculate the area of the circle

The diameter of the circle is equal to the diagonal of the square. Given the side length of the square \(a = 10\) cm. Using the Pythagorean theorem for the square (where \(d\) is the diagonal), \(d=\sqrt{10^{2}+10^{2}}=\sqrt{200} = 10\sqrt{2}\) cm. The radius of the circle \(r=\frac{d}{2}=5\sqrt{2}\) cm. The area of the circle \(A_{circle}=\pi r^{2}=\pi\times(5\sqrt{2})^{2}=50\pi\approx50\times3.14 = 157\) \(cm^{2}\).

Step2: Calculate the area of the square

The area of the square \(A_{square}=a^{2}\), with \(a = 10\) cm, so \(A_{square}=10\times10=100\) \(cm^{2}\).

Step3: Calculate the area of the shaded region

The area of the shaded region \(A = A_{circle}-A_{square}\). Substitute the values: \(A=157 - 100=57\) \(cm^{2}\).

Answer:

57