Question

inscribe a rectangle of base b and height h in a circle of radius one, and inscribe an isosceles triangle in a region of the circle cut off by one base of the rectangle (with that side as the base of the triangle). for what value of h do the rectangle and triangle have the same area?

Answer 1

let the height of the triangle be y. Then since the center of the rectangle is also the center of the circle,
y + h/2 = 1
and so we have
bh = by/2
bh = b(1-h/2)/2
solve for h, and use (b/2)^2 + (h/2)^2 = 1
to find b