# What is the area percentage of a ⌈square in a circle? What is the area percentage of a ⌈circle in a square?

Useless Backstory:

I was trying to guess how many jellybeans were in a jar. If the jar was a glass cube or a glass box, I would just count width × length × height. With a cylinder however, I figure I could still use this method, but just minus the ratio of circle in square. So I became curious to see if the ratios were the same when reversed.

Take a circle, any radius. Now grow a square inside of it, until the four corners of the square touch the perimeters of the circle.

Will the area of the square always hold the same ratio to the area of the circle? If yes, what is that ratio?

Now vice versa

Take a square, any radius. Now grow a circle inside of it, until the circle perimeter touches each side of the square.

Will the area of the circle always hold the same ratio to the area of the square? If yes, what is that ratio?

The circle of radius R in the square of side C is such :

$2 R = C$

Area of the circle $\pi R^2$ , of the square $C^2$ , and the ratio of circle on square is :

and the ratio is : $\pi \frac{R^2}{C^2} = \pi \frac{R^2}{4 R^2} = \frac{\pi}{4}$

If the area of the square is $100 cm^2$ , the circle will have an area of $\frac{\pi}{4} 100 cm^2 \approx 78.54 cm^2$

The square of side C in the circle of radius R is such :

$C^2 = R^2 + R^2$ similarly the ratio is : $\pi \frac{R^2}{C^2} = \pi \frac{R^2}{2 R^2} = \frac{\pi}{2}$

If the area of the square is $100 cm^2$ , the circle will have an area of $\frac{\pi}{2} 100 cm^2 \approx 157,08 cm^2$

• By "rayon" I think you mean "radius".
– user856
Aug 19 '16 at 4:52
• Keep in mind, the same basic principle applies, but the ratios change in 3 dimensions. Look up "Packing Fraction" for more detail on these concepts. Sep 14 '18 at 15:52