Netherlands
Find the area of the square that can be inscribed in a circle of radius ^@6 \space cm^@.
Answer:
^@ 72 \space cm^2 ^@
- 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^@.
