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A circle circumscribed a rectangle with sides 16cm and 12cm. What is the area of the circle?
A. \[48\pi \]square cm.
B. \[50\pi \]square cm.
C. \[100\pi \]square cm.
D. \[200\pi \]square cm.

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Last updated date: 14th Jul 2024
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Answer
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Hint: In this problem, we are given that a circle circumscribed a rectangle with sides 16cm and 12cm, we have to find the area of the circle. We can first draw a suitable diagram for the given measurements. We know that area of the circle formula is \[\pi {{r}^{2}}\], where r is the radius. We can find the value of r where We can see that the diagonal of the rectangle will be the diameter of the circle. Using Pythagoras theorem we can find the radius value and substitute in the area formula to get the area value.

Complete step by step answer:
Here we have to find the area of a circle which circumscribes a rectangle with sides 16 cm and 12cm.
We can now draw the diagram.
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We can see that the diagonal of the rectangle = the diameter of the circle = 2r.
We can now find the diagonal of the rectangle using the Pythagoras formula,
\[\Rightarrow {{\left( 2r \right)}^{2}}={{\left( 16 \right)}^{2}}+{{\left( 12 \right)}^{2}}\]
We can now simplify the above step, we get
\[\begin{align}
  & \Rightarrow 4{{r}^{2}}=256+144=400 \\
 & \Rightarrow {{r}^{2}}=100 \\
 & \Rightarrow r=10cm \\
\end{align}\]
The radius of the circle is 10cm.
We can now substitute the radius in the formula for area of circle \[\pi {{r}^{2}}\], we get
Area of the circle = \[\pi \times {{\left( 10 \right)}^{2}}=100\pi c{{m}^{2}}\].
So, the correct answer is “Option C”.

Note: We should know that if a circle circumscribes a rectangle, then the diameter of the circle is the diagonal of the rectangle, where the diagonal of the rectangle = the diameter of the circle = 2r. We should remember that, the formula to find the area of the circle is \[\pi {{r}^{2}}\].