To find the surface area of any prism, including cylinders because a cylinder is just a prism with circles for bases (take a regular sheet of paper which is a rectangle and roll it up and what do you get?), I teach my 7th grade students a generic formula: S=2B + Ph
B=area of 1 base of the prism
P=perimeter of 1 base of the prism
h=the height of the prism
First, let's make sure we understand what each of these parts are and how to identify them. A prism is a polyhedron (3 dimensional shape) that is made from connecting two congruent polygons (a 2 dimensional shape like a triangle or rectangle) bases together using 1 or more rectangles. A simple example of a common prism seen all the time is a box. A box has a top and a bottom that are the same size and shape, and all the sides of the box that connect that top and bottom are rectangles. So in this example of the box, the base would be either the top or bottom of the box. It really doesn't matter if you use the top or bottom, because they are identical or congruent. In a cylinder, the bases would be the circular ends, because those are the top and bottom that are congruent. In a triangular prism, the triangle ends that are the same would be your bases.
The height of a prism is the length of the line that connects the congruent bases to each other. In the example of the box, the height would be the length of the edge that connects the top and bottom of the box.
So now that the vocabulary is defined, let's use the formula S=2B+Ph to find the surface area of a box that has a base that is 4in by 3in, and a height of 6in. First we find the area of the base or B. For a rectangle, that is length times width, so in this case it is 4x3 or 12in squared.
Next we find the perimeter of the base or P. Perimeter means all around the outside of a shape, so it just means add up the sides of the rectangle that is the base. In our case, it would be 4+4+3+3 or 14 inches.
Now we need the height of the prism. Make sure not to confuse the height of the prism with other heights that occur in geometry, like the height of a triangle for example. Remember, the height of a prism is the length of the line that connects 1 base to the other base. Our height is 6in as stated in the beginning of the problem.
Now that we have all the necessary numbers, let's plug them in and simplify:
S=108in squared (area is always units squared)
With a triangular prism, for the B, or base area, you would use the formula for the area of a triangle, or a=1/2bh (1/2 of the base of the triangle times the height of the triangle). Remember though, the h in this formula is not the height of the prism; rather it is the height of the triangle base, which is a perpendicular line (90 degrees) from the base of your triangle to the top of your triangle. To help my students avoid confusion with this, I have them draw the triangle to the side as a simple polygon, label the parts of it, and find its area. That will then serve as B in the S=2B+Ph. The P would just be the sum of the length of the 3 sides of the triangle base. The height of the prism in S=2B+Ph is the length of the line that connects the top triangle to the bottom triangle.
For a cylinder, since the bases are circles, you would use (pi*r)squared to find B, and 2*pi*r to find P. The height would just be the length of the line that connects the two circular bases.
I hope that this helps Tuty, good luck.