Find the area of the square that can be inscribed in a circle of | Math Question and Answer

Answer:

$72 \space cm^2$

Step by Step Explanation:

Let us draw a circle with $O$ as a center and radius $6 \space cm$ .
Now, let us draw a square $ABCD$ inside the circle. We see that the diagonal of the square is equal to the diameter of the circle.

Radius of the circle = $6 \space cm$
So, diameter of the circle $= 2 \times$ Radius $= 2 \times 6 \space cm = 12 \space cm$
Therefore, diagonal of the square $= 12 \space cm$
Using Pythagores' theorem in $\triangle ABC$ , we have $\begin{aligned} & AB^2 + BC^2 = AC^2 \\ \implies & AB^2 + BC^2 = (12)^2 && \text { [AC is the diagonal of the square.] } \\ \implies & AB^2 + AB^2 = 144 && \text { [Sides of a square are equal.] } \\ \implies & 2 AB^2 = 144 \\ \implies & AB^2 = 72 \end{aligned}$
We know, $\begin{aligned} & \text { Area of the square } = Side^2 \\ \implies & \text { Area of the square } = AB^2 = 72 \space cm^2 \end{aligned}$
Thus, the area of the square that can be inscribed in a circle of radius $6 \space cm$ is $72 \space cm^2$ .

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