
Solids Produced by Rotating Polygons
Another type of Math IC question that you may come across
involves a solid produced by the rotation of a polygon. The best
way to explain how this type of problem works is to provide a sample
question:

When this triangle is rotated about AB,
a cone is formed. To solve the problem, the first thing you should
do is sketch the cone that the triangle will form.
The question asks you to figure out the surface area of
the cone. The formula for surface area is πr^{2} +
πrl, which means you need
to know the lateral height of the cone and the radius of the circle.
If you’ve drawn your cone correctly, you should see that the lateral
height is equal to the hypotenuse of the triangle. The radius of
the circle is equal to side BC of the
triangle. You can easily calculate the length of BC since
the triangle is a 306090 triangle. If the hypotenuse is 2, then BC,
being the side opposite the 30º angle, must be 1. Now plug
both values of l and r into
the surface area formula and then simplify:
Common Rotations
You don’t need to learn any new techniques or
formulas for problems that deal with rotating figures. You just
have to be able to visualize the rotation as it’s described and
be aware of which parts of the polygons become which parts of the
geometric solid. Below is a summary of which polygons, when rotated
a specific way, produce which solids.



A rectangle rotated about its edge produces a cylinder.  A semicircle rotated about its diameter produces a sphere.  A right triangle rotated about one of its legs produces a cone. 



A rectangle rotated about a central axis (which must contain the midpoints of both of the sides that it intersects) produces a cylinder.  A circle rotated about its diameter produces a sphere.  An isosceles triangle rotated about its axis of symmetry (the altitude from the vertex of the noncongruent angle) produces a cone. 
