Questions about area

Question

In math class (I'm in Geometry) I was messing around and decided to try and find the area of a circle using the area of a square if the radii are the same length.

The square is inscribed in the circle.

I got $\displaystyle\left(\frac n2\right)\pi=A$ where $n$ is the area of the square (I can show my work if needed).

I've also tried with triangles and pentagons but I'm unsure that they are right so I guess my question is:

Can you extend something like this into 3D like with a cube and sphere?

Also I can show my work if you guys want to see if its right.

Answer 1

Why not?

You could use exactly same argument as in 2D case, but this time noting that the expression for the radius of the circle (in terms of volume of cube) is rather different from that in 2D.

Rather more interesting thing to do would be calculating area of regular n-gon inscribed in a circle and how the limit of area behaves as n approaches $\infty$ or how limit of area of regular n-gons circumscribing a circle behaves.

Answer 2

If $n$ represents the area of a square as you say, then $\sqrt{n}$ represents a side of the square. Now take half of that side (which is $\frac {\sqrt{n}}{2}$).

See the above picture: Recognize from the 45-45-90 triangle that the diagonal of the triangle--which is also the radius of a circle that circumscribes a square--will be $r=\frac {\sqrt{n}}{2} \sqrt{2}$. Your area of the circle should be $$A=\pi r^2=\pi \left(\frac {\sqrt{n}}{2} \sqrt{2} \right)^2=\frac{2 \pi n}{4}=\frac{\pi n}{2}$$where, again, $n$ denotes the area of the square.