A circle has an inscribed square. The diagonal of the square is 21 cm. Calculate the area of the shaded…

Question

A circle has an inscribed square. The diagonal of the square is 21 cm. Calculate the area of the shaded region between the circle and the square. Use pi = 22/7.

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The figure shows a circle with an inscribed square. The shaded part is the area of the circle minus the area of the inscribed square.

Identify the dimensions of the circle and the square.
The given length of 21 cm is the diagonal of the inscribed square and also the diameter of the circle.
So, the diameter of the circle $d = 21$ cm.
The radius of the circle $r = \frac{d}{2} = \frac{21}{2}$ cm.
For the inscribed square, its diagonal $D = 21$ cm.
We are given $\pi = \frac{22}{7}$ .
Calculate the area of the circle.
The formula for the area of a circle is $A_c = \pi r^2$ .
$A_c = \frac{22}{7} \times \left(\frac{21}{2}\right)^2$ $A_c = \frac{22}{7} \times \frac{21 \times 21}{2 \times 2}$ $A_c = \frac{22}{7} \times \frac{441}{4}$ $A_c = \frac{22 \times 441}{28}$ $A_c = \frac{11 \times 441}{14}$ $A_c = \frac{4851}{14}$ $A_c = 346.5 \text{ cm}^2$
Calculate the area of the inscribed square.
The formula for the area of a square using its diagonal $D$ is $A_s = \frac{1}{2}D^2$ .
$A_s = \frac{1}{2} (21)^2$ $A_s = \frac{1}{2} (441)$ $A_s = \frac{441}{2}$ $A_s = 220.5 \text{ cm}^2$
Calculate the area of the shaded part.
The area of the shaded part is the area of the circle minus the area of the square.
$\text{Shaded Area} = A_c - A_s$ $\text{Shaded Area} = 346.5 \text{ cm}^2 - 220.5 \text{ cm}^2$ $\text{Shaded Area} = 126 \text{ cm}^2$
The area of the shaded part is $\text{126 cm}^2$ .

A circle has an inscribed square. The diagonal of the square is 21 cm. Calculate the area of the shaded region between the circle and the square. Use pi = 22/7.

Answer

Step-by-step solution

Identify the dimensions of the circle and the square.

Calculate the area of the circle.

Calculate the area of the inscribed square.

Calculate the area of the shaded part.

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