A chord of a unit circle creates a segment with an area of one. What is the length of the chord?

Question

You have a circular cake with a radius of one foot. You intend to cut this cake with one straight slice to create a piece with an area of one square foot. What must the length of the cut be?

 

Answer 1

The diameter of the cake is a chord.

The area of the semicircle is (1/2)π(1)² = π/2 square feet

 

So we want a chord parallel to the diameter to create a semicircle with area of 1 square foot

 

Area of semicircle = 1 sq ft = (1/2)π(r)²

 

(2/π) = r²

 

r = √(2/π)

 

The chord of the original cake is the diameter for the semicircle so the chord is 2r = 2√(2/π)

Answer 2

I haven't done this in a while so it would probably confuse us both if I tried to explain it, but here's a website that may help: http://mathworld.wolfram.com/CircularSegment.html