Solving the sample problem

A glass prism is an equilateral triangle on one end and is 6 inches long. The perimeter of the triangle is 3 inches. What is the total surface are of the prism, exactly?

Well, let's see... we have a triangle with perimeter of 3", so each side of the triangle must be one inch. We are trying to figure out the total surface area which includes 3 large rectangles and 2 small triangles (on at each end).
The rectangles area easy 6"x 1" = 6 sq. inch per rectangle times 3 rectangles is 18 sq. in.
The triangles have a base of one and a height of ... hm. Draw the red line shown in the diagram. We now have a right triangle with legs of 1/2 and h, and a hypotenuse of 1. Turn to Pythagorus for help: a² + b² = c² .
(½)² + h²= 1²

¼+ h²= 1

h²= ¾ (Now take the square root of both sides which means taking the square root of both the numerator and denominator on the right hand side to get ...

So our total area is 18 plus 2 * 1/2 bh Hence 18 + bh, or 18 +( 1) √3 / 2

sq in