Question

circle o is inscribed in square efgh, as shown below.
e f
h g
the circumference of circle o is 20 centimeters. which of the following is closest to the perimeter of square efgh?
25.5 cm
27 cm
28.5 cm
24 cm

Explanation:

Step1: Recall the circumference formula of a circle

The formula for the circumference of a circle is \( C = \pi d \), where \( C \) is the circumference and \( d \) is the diameter. We know \( C = 20 \) cm, so we can solve for \( d \): \( d=\frac{C}{\pi}=\frac{20}{\pi} \) cm.

Step2: Determine the side length of the square

Since the circle is inscribed in the square, the diameter of the circle is equal to the side length \( s \) of the square. So \( s = \frac{20}{\pi} \) cm.

Step3: Calculate the perimeter of the square

The perimeter \( P \) of a square is given by \( P = 4s \). Substituting \( s=\frac{20}{\pi} \) into the formula, we get \( P = 4\times\frac{20}{\pi}=\frac{80}{\pi}\approx\frac{80}{3.14}\approx25.5 \) cm? Wait, no, wait, \( \pi\approx3.14 \), \( \frac{20}{\pi}\approx6.37 \), then \( 4\times6.37 = 25.48\approx25.5 \)? Wait, but let's check again. Wait, the circumference is \( C = 2\pi r \), but diameter \( d = 2r \), so \( C=\pi d \), so \( d = C/\pi = 20/\pi\approx6.37 \) cm. Then the side of the square is equal to the diameter, so perimeter is \( 4\times6.37\approx25.48\approx25.5 \) cm? But wait, the options are 25.5, 27, 28.5, 24. Wait, maybe I made a mistake. Wait, no, let's recalculate. \( \pi\approx3.1416 \), so \( 20\div3.1416\approx6.366 \), then \( 4\times6.366\approx25.46\approx25.5 \) cm. So the closest is 25.5 cm.

Answer:

25.5 cm