Logic Puzzles

Could you show your working on how you derive the multiple of pi that you the area from the chord length. I've run out of room on the paper bag that I'm doodling on.

Could you show your working on how you derive the multiple of pi that you the area from the chord length. I've run out of room on the paper bag that I'm doodling on.

If the chord (c) is just the diameter of another circle with the same area, you get

A=π r^2 where r=c/2, so
A=π (c/2)^2 = π/4 c^2

Note there's no radius, of either circle, in the result.

The long way is to take the area:

A=π R^2 - π r^2
A=π (R^2 - r^2)

and find a way to relate the two radii to the chord length.
So, draw the donut, with the chord of unknown length c. Then draw a radius length r of the inner circle, meeting the chord in the middle at a right angle, and a radius length R of the outer circle, meeting the end of the chord

You now have a right triangle, with hypotenuse R and other sides r and c/2, so

R^2 = r^2 + (c^2)/4
R^2 - r^2 = (c^2)/4

substitute this into the area equation above to get

A=π (c^2)/4