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प्रश्न
If a square is inscribed in a circle, what is the ratio of the areas of the circle and the square?
उत्तर
We have the following situation
Let BD be the diameter and diagonal of the circle and the square respectively.
We know that area of the circle =`pir^2`
Area of the square = `"side"^2`
As we know that diagonal of the square is the diameter of the square.
Diagonal=`2r`
Side of the square= `"diagonal"/sqrt2............(1)`
Substituting "diagonal"=2r in equation (1) we get,
side of the square=`(2r)/sqrt2`
Now we will find the ratio of the areas of circle and square.
Area of circle/Area of square=`(pir^2)/((2r)/sqrt2)^2`
Now we will simplify the above equation as below,
Area of circle/Area of square=`(pir^2)/(r^2/2)`
Area of circle/Area of square=`pir^2xx2/(4r^2)`
Hence, `"Area of circle"/"Area of square"-pi/2`
Therefore, ratio of areas of circle and square is `pi:2`
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