Question

Shown below is a cross-section which is revolved 270 degrees around the P-Q axis to create a solid. How many surfaces will the resultant solid have?

Answer 1

Correct Answer: 10

Step 1: Split the boundary into the 5 arc outer profile and the single hole loop.

Step 2: These are closed features, so revolving them at any angle gives 5 outer curved surfaces plus 1 hole cylinder, 6 surfaces in total.

Step 3: Stopping the sweep at 270 degrees instead of 360 opens up two flat cut faces where the material starts and ends.

Step 4: Each cut face splits into an outer flat part and an inner flat part around the hole, adding 4 more surfaces. \[ \boxed{10} \]

Answer 2

A good check is to first imagine the same cross section revolved a full 360 degrees, then see what changes when the sweep is cut short at 270 degrees instead.

A full 360 degree revolve would close up completely, leaving only the curved surfaces: 5 from the outer profile and 1 from the hole, 6 surfaces with no flat ends needed. Stopping at 270 degrees instead removes that closure and opens up two flat cut faces where the sweep begins and ends. Because the section has a hole, each cut face is really two flat pieces, an outer piece and an inner piece around the hole, so the two cut faces contribute 4 flat surfaces. Adding these 4 flat surfaces to the 6 curved ones from the full-revolve case gives the same total as before.

the answer is 10