Largest Rectangle inside a Circle

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Let's draw a circle with a radius of r, and install a rectangle inside it.

From the video, the 2 angles α and β have a total of 180 degrees.

In this article, we are going to prove that the largest rectangle it can have inside is actually a square.

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by using the cosine rule, we can know that

and

therefore, to get the area of the rectangle, we can have

to make sure that the area has a maximum value, we just need to differentiate this equation twice and see if the result is negative or not.

since we are dealing with 0° ≤ α ≤ 180°, then sinα > 0, then -2r²sinα < 0, therefore d²A/dα² < 0, meaning the Area of the rectangle has a valid maximum value.

If we plot a graph for A and α, when A reaches the maximum value, dA/dα = 0

from this we can know that

when

and when α = 90°, then β = 90°

the whole shape inside the circle is a square.

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Do you know ?

This proof is inspired by the main task of filling up a circular region in Minecraft by using the command /fill in order to achieve the maximum efficiency