Question

The ratio of the areas of the incircle and the circumcircle of a square

Answer 1

Answer:

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The incircle of a square touches all the four sides. The diameter of the circle is equal to the length of the square's side. The radius of the circle is L/2. The circumcircle of a square touches all the four vertices of the circle.

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Answer 2

Answer is 2:1.

EXPLAINATION

.Let us assume that the side of the square is "x", Radius of incircle is "r" and radius of circumcircle is "m".

Since the side of square is equal to the diameter of incircle, the side will be half of radius.

r = x/2-------- eq 1

We know that the diagonal of a square is equal to "side root 2"------------ eq 2

Therefore, by extending the radius of circumcircle such that, it touches the vertices of the square. We get the diagonal of the square and also the diameter of the circumcircle.

Using equation 2 we get,

$m = x \sqrt{2}$

Now.

area of incircle/area of circumcircle =

$\pi \: r {}^{2} \div \pi \: m {}^{2}$

which gives. 2:1