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?

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

Oh thank you very much sir!!

To solve this problem, we need to find the height of the cylinder that is formed when the rhombus is rolled up.

Let's first find the base area of the cylinder. Since the side length of the rhombus is 6, the length of AB (or DC) is also 6. The perimeter of the base of the cylinder is equal to the circumference of the rhombus, which can be calculated by multiplying the length of AB by 4. So, the perimeter of the base is 4 * 6 = 24.

We know that the volume of the cylinder is given as 6, and the formula for the volume of a cylinder is V = base area * height.

Let's solve for the base area:
V = base area * height
6 = base area * height

Since we know that the perimeter of the base is 24, and the base of the cylinder is a circle, we can calculate the radius of the base (r) using the formula for the circumference of a circle: C = 2πr. Here, the circumference C is equal to 24. Solving for r, we have:
24 = 2πr
r = 24 / (2π)
r = 12 / π

Now, we can find the base area (A) using the formula for the area of a circle: A = πr^2.
A = π * (12/π)^2
A = π * (12^2 / π^2)
A = 144/π

Now that we have the base area and volume, we can solve for the height (h):
6 = (144/π) * h
h = (6 * π)/144
h = π/24

To find sine of angle ABC, we need to find the ratio of the height and side length:
sin(angle ABC) = h/6
sin(angle ABC) = (π/24) / 6
sin(angle ABC) = π/144

So, sin(angle ABC) is equal to π/144.