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# A wire when bent in the form of a square encloses an area of $992.25\text{ c}{{\text{m}}^{2}}$ . If the same wire is bent in the form of a semicircle, what will be the radius of the semicircle so formed?

Last updated date: 25th Jun 2024
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Hint: To find the radius of the semicircle formed from a wire that is used to make a square, we will find the side of the square using the given area and equation $\text{Area}={{a}^{2}}$ . Since the same wire is used to make a semicircle, we know that the perimeter of the square = circumference of the semicircle. Let us use the formula for circumference of semicircle, that is, $\pi r+2r$ and substitute this to the perimeter of the square to get the value of the radius.

We need to find the radius of the semicircle obtained. It is given that the wire used to create the semicircle was bent in the form of a square with an area $992.25\text{ c}{{\text{m}}^{2}}$ .
Hence, we can write the area of square of side a as
$\text{Area}={{a}^{2}}$

We know that the given area is $992.25\text{ c}{{\text{m}}^{2}}$ .Hence,
${{a}^{2}}=992.25$
Now, let us take the square root to get the side of the square.
$a=\sqrt{992.25}=31.5\text{cm}$
Let us find the perimeter of the square.
We know that perimeter of a square is given by
$\text{Perimeter}=4a$
Let us now substitute the values. We will get
$\text{Perimeter}=4\times 31.5=126\text{cm}$
It is given that the same wire is used to form a semicircle. Hence,
$\text{Perimeter of the square }=\text{ circumference of the semicircle…}\left( \text{i} \right)$
We know that the circumference of a semicircle with radius r is given as
$\text{Circumference}=\pi r+2r$
Let us substitute this in (i). We will get
$\pi r+2r=126$
Let us take r outside from LHS. We will get
$r\left( \pi +2 \right)=126$
We can take the value of $\pi =\dfrac{22}{7}$ . Hence, the above equation becomes
$r\left( \dfrac{22}{7}+2 \right)=126$
Let us solve us. We will get
$r\left( \dfrac{36}{7} \right)=126$
Taking the constants to one side, we will get
$r=126\times \dfrac{7}{36}$
We can now solve this. We will get
$r=24.5\text{cm}$

Hence, the radius of the semicircle is 24.5 cm.

Note: You must know the formulas of different shapes starting from perimeter to the area. You may mistake the area of square as 4a and the perimeter of the square as ${{a}^{2}}$ . You may use the formula for circumference of a circle , that is, $2\pi r$ instead of $\pi r+2r$ .