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# Geometry

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A square is inscribed in a circle forming 4 identical segments. The area of a single segment is 1cm2. What's the area of that square?

Dec 14, 2019

#1
+107089
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A square is inscribed in a circle forming 4 identical segments. The area of a single segment is 1cm2. What's the area of that square?

$$\text{ Area of square is }\frac{1}{2}*2r*2r=2r^2 \\ \text{ This formula is gained treating the square as a special case of a kite.} \\ \pi r^2- 2r^2=4 \\ r^2(\pi -1)=4 \\ r^2=\frac{4}{\pi-1}\\ \text{So area of the square is }\frac{8}{\pi-1}\;\;cm^2$$

Take 2$$\text{ Area of square is }\frac{1}{2}*2r*2r=2r^2 \\ \text{ This formula is gained treating the square as a special case of a kite.} \\ \pi r^2- 2r^2=4 \\ r^2(\pi -\color{red}2\color{black})=4 \\ r^2=\frac{4}{\pi-2}\\ \text{So area of the square is }\frac{8}{\pi-2}\;\;cm^2$$

This time I checked my answer

The area of the square is approx 7 and the area of the circle is approx 11 so that is good.

Still be careful to check my working very carefully, there is always room for careless errors.

code:

\text{ Area of square is  }\frac{1}{2}*2r*2r=2r^2 \\ \text{ This formula is gained treating the square as a special case of a kite.}       \\

\pi r^2- 2r^2=4 \\
r^2(\pi -\color[red]{2})=4 \\
r^2=\frac{4}{\pi-2}\\
\text{So area of the square is  }\frac{8}{\pi-2}\;\;cm^2

Dec 14, 2019
edited by Melody  Dec 14, 2019
edited by Melody  Dec 14, 2019
#2
+180
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Hi, Melody!

According to your calculations, the area of a square is smaller then the combined area of all 4 segments.

When you inscribe a square in a circle, it takes up about 64% of circle's area.

Btw, I've created this problem.

Dec 14, 2019
edited by Dragan  Dec 14, 2019
#3
+107089
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Yea I made a stupid mistake with factoizing.  I have fixed it now.

Was this intended to be some kind of test for me or what?

You did tell me I was patient, it seems  I passed that test for the second time.

Melody  Dec 14, 2019
edited by Melody  Dec 14, 2019