Just Differentiate It!

Geometry Level 1

A square is inscribed in a circle . If the area of the circle is π 2 \pi^2 , what is the area of the square?

π \pi 2 π 2\pi 3 π 3\pi 4 π 4\pi

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Pi Han Goh by Pi Han Goh , 30, Malaysia

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3 solutions

Since area if circle is π 2 π^2 , radius of the circle is π \sqrt{π} .

Diameter of circle = Diagonal of square = 2 π 2\sqrt{π} .

Therefore side of the square is 2 π 2 = 2 π \dfrac{2\sqrt{π}}{\sqrt2}=\sqrt{2π} .

Therefore area of square is 2 π 2π .

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Moderator note:

The diagonal of the square could be found by applying Pythagorean Theorem ,

( \side length ) 2 + ( \side length ) 2 = ( diagonal of square ) 2 = ( diameter of circle ) 2 . (\text{\side length})^2 + (\text{\side length})^2 = (\text{diagonal of square})^2 = (\text{diameter of circle})^2 .

Since we know that the diameter of the circle is twice its radius, then 2 ( side length ) 2 = ( 2 π ) 2 2(\text{side length})^2 = (2 \sqrt\pi)^2 .

And because we know that the area of a square is simply the square of its side length, then the answer is 1 2 ( 2 π ) 2 = 2 π \dfrac12 (2\sqrt\pi)^2 = 2\pi .

Andrea Gallese
Aug 11, 2016

Just differentiate d π 2 d π = 2 π \frac{d \pi ^2}{d \pi} = 2\pi

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please explain how differentiating gives us answer ??

Chirayu Bhardwaj - 4 years, 10 months ago

Im whith chiravu

Singal Parth - 4 years, 10 months ago

Please explain

Anirudha Brahma - 4 years, 10 months ago
Md Zuhair
Aug 4, 2016

Here we see that A = π 2 A = \pi^2 And we also know that π r 2 = π 2 \pi r^2 = \pi^2 .... Or r = π r = \sqrt{\pi} ... and A 2 = π \frac{A}{\sqrt{2}} = \sqrt{\pi} .. Solving for a 2 a^2 we get 2 π 2\pi

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