Question

Rhombus ABCD, with side length 6, is rolled to form a cylinder of volume 6 by taping AB to DC. What is sin(angle ABC)?

I don't know how to begin. Help?

Answer 1

Start by drawing a reasonably accurate rhombus on a piece of paper.
Remember a rhombus has 4 equal sides and opposite angles equal, ( A square is a special kind of rhombus, so don't use a square)

cut your rhombus out with scissors.
Surprisingly, you will be able to form a cylinder, note that the circumference at both ends equals the side of the rhombus.
BUT, the height of the cylinder is NOT equal to the side of the rhombus.
We need the radius of the circular ends
we know 2πr = 6
r = 3/π
The circular ends have area π(3/π)^2 = 9/π
let the height be h
so we know:
πr^2 h = 6
π(9/π)h = 6
h = 6/9 = 2π/3

unfold your paper and draw a right-angled triangle at the acute angle,
so sin Ø = (2π/3) / 6
= π/9
Ø = 20.43°

check my arithmetic

Answer 2

Oh thank you very much sir!!