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In a Spin

What is the volume of the solid formed by rotating this right angled triangle about the hypotenuse?

Canny Fraction

Stage: 4 Short Challenge Level: Challenge Level:1

Diagram with cylinder and volumes of liquid shown

Imagine the liquid became solidified.
Rotate the can until the face which contains XY is horizontal.
Now take the horizontal cross-section half way up the can which meets the plane face of the "liquid" in the line AB.
Observing that AB bisects the radius PQ, rotate the upper section of the "liquid" about AB through 180 degrees. The new position is shaded darkly and fits into the bottom right hand quarter (with some space to spare).
The actual fraction can be shown to be approximately 0.21. The answer is therefore less than a quarter.

This problem is taken from the UKMT Mathematical Challenges.
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