Question

The given figure shows a square of side 2a. Find the ratio between the areas of the incircle and the circum-circle of the square.

Answer 1

Hint: Let us find the area of the incircle and the circumcircle of the square and divide them to find the required ratio.

Complete step-by-step answer:
As we know that incircle to a square is the circle inside the square.
And, the circumcircle is the circle inside which the square is inscribed.
As we can see from the above given figure that the diameter of the circumcircle is the diagonal of the given square (i.e. CA).
As we know that the side of the square is 2a.
And all angles of the square are \[{90^0}\](\[\angle {\text{ABC}} = {90^0}\])
So, according to the Pythagorean theorem we know that in a right-angle triangle XYZ, which is right-angled at Y. \[{\text{X}}{{\text{Y}}^2} + {\text{Y}}{{\text{Z}}^2} = {\text{Z}}{{\text{X}}^2}\]
So, according to Pythagorean theorem in \[\Delta {\text{ABC}}\],
\[{\text{A}}{{\text{B}}^2} + {\text{B}}{{\text{C}}^2} = {\text{C}}{{\text{A}}^2}\]
\[{\left( {{\text{2a}}} \right)^2} + {\left( {{\text{2a}}} \right)^2} = {\text{C}}{{\text{A}}^2}\]
So, CA = \[2{\text{a}}\sqrt 2 \](diameter of the circumcircle)
So, the radius of the circumcircle will be OA = \[\dfrac{{{\text{CA}}}}{2} = {\text{a}}\sqrt 2 \]
Now we had to find the radius of the incircle.
As we can see from the above figure that the radius of the incircle is OE.
And we know that the circle is inside a square, and all sides of the square are the same. So, due to the symmetry centre of the incircle should be the same as the centre of the square.
As we know that line drawn perpendicular from the centre of the circle to any of its sides, bisect its side from the middle point. So, AE = EB = a.
And as we know that the perpendicular distance from the centre of the square to any of its sides is half of the side length of the square.So, OE = a.
So, the radius of the incircle will be OE = a
As we know that the area of a circle having radius r is given as \[\pi {{\text{r}}^2}\].
So, ratio of area of incircle to the area of the circumcircle will be \[\dfrac{{\pi {{\left( {{\text{OE}}} \right)}^2}}}{{\pi {{\left( {{\text{OA}}} \right)}^2}}}\]
So, ratio will be \[\dfrac{{\pi {{\left( {\text{a}} \right)}^2}}}{{\pi {{\left( {{\text{a}}\sqrt 2 } \right)}^2}}} = \dfrac{{{{\text{a}}^2}}}{{2{{\text{a}}^2}}} = \dfrac{1}{2}\].
Hence, the ratio of the area of the incircle and circumcircle to the given square will be \[\dfrac{1}{2}\].

Note: Whenever we come up with this type of problem then first, we have to find the diagonal length of the given square by using the Pythagorean theorem because the diagonal of the square is also the diameter of the circumcircle. After that we find the radius of the incircle by using the theorem which states that the perpendicular line drawn from the centre of the square to any of its side has length equal to the half of the side of the square. And then we had to find the area of both circles by the formula of area of circle which states that if the radius of the r then its area will be \[\pi {r^2}\]. After that divide both areas to get the required ratio.