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A wire of 5024 m length is in the form of a square. It is cut and made a circle. Find the ratio of the area of the square to that of the circle.

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Last updated date: 21st Jul 2024
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Answer
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Hint: We need to be aware about the formula for perimeter and area of both square and that of a circle. Here, a wire is given in the form of a square i.e. the length of the wire will be its perimeter. Using this information, we could calculate the side of the square. After calculating the side we can use the formula for the circumference of a circle and equate it with the same length and find the radius of the circle. After radius and side have been calculated, the ratio of their areas is easy to calculate.

Complete step by step answer:
We are given a wire that has been molded in the shape of a square which means that the perimeter of the square is the length of the wire. The perimeter of a square of side a is $4a$, so we have:
$4a=5024$m
$\implies a=1256$m
Now, the same wire has been converted into a circle. The circumference of a circle of radius $r$ is $2\pi r$, and this will be equal to the length of the wire. So we have:
$2\pi r=5024$m
$\implies \pi r=2512$m
$\implies r=\dfrac{2512}{\pi}$
Now, we need to calculate the ratios of the areas.
Area of the square is given by $a^2$.
Area of a circle is given by $\pi r^2$
So, we need to find:
$\dfrac{{{1256}^{2}}}{\pi \times {{\left( \dfrac{2512}{\pi } \right)}^{2}}}$
$=\dfrac{1256\times 1256\times 22}{2512\times 2512\times 7}$
$=\dfrac{11}{7\times 2}=\dfrac{11}{14}$
Hence, the required ratio is 11:14.

Note: If you are not aware about the formulae related to the area and perimeter of a square and circle then you will not be able to solve this question, so it should be known. Moreover, do not try to square such big terms because later on, the ratio is to be found out which means that some terms will automatically get cancelled. So to make least calculation mistakes, do not square the terms prior to finding the ratio.