The surface area of a three-dimensional object is the total area of all the sides that make up the object. For example, a cube is made of six squares, a right triangular prism is made of three rectangles and two triangles, and a cylinder is made of two circles and a rectangle (wrapped around, instead of laying flat).
Cube: The area of a square is the length of one side squared. Since the cube is made of six of these squares, multiply the result by six.
Example: A 3 inch cube has surface area
32 × 6 = 54 square inches.
Right Triangular Prism: The area of a triangle is 1/2 the base of the triangle times the height of the triangle. The area of a rectangle is the length of one side times the length of an adjacent side. A Right Triangular Prism is made of two triangles and three rectangles, so multiply the area of the triangle by two and add the areas of the three rectangles.
Example: A 7" tall right prism with a 3" by 4" by 5" right triangle base has surface area
2 × (3 × 4) + (3 × 7) + (4 × 7) + (5 × 7) = 24 + 21 + 28 + 35 = 108 square inches.
Cylinder: The area of a circle is π × r2 and the area of a rectangle is the length of one side times the length of an adjacent side. In this case, the length of one side of the rectangle is the same as the circumference of the circle (given by π × d), and the length of the other side is the height of the cylinder. Multiply the area of the circle by two and add the area of the rectangle.
Example: A 5" tall cylinder with ends of a 4" diameter has surface area
2 × π × (4/2)2 + π × 5 = 25.1 + 15.7 = 40.8 square inches.
...and so forth. All we need to do is add up all the areas of all the sides of the object. But what if the object isn't made of nice flat sides? For example, how do we find the surface area of a sphere? The long answer involves calculus, and integrating to find the surface of a revolution about an axis, but for the purposes of this writeup I will assume no experience with advanced mathematics. The surface area of many common shapes has been found for us and all we really have to do is look up the formula:
4 × π × r2.
Likewise the surface area of a cone is not easily found without calculus. Its formula is:
π × r2 + π × r × s
Where r is the radius of the circle making up the base of the cone and s is the length of the side of the cone from the edge of the circle to the tip. It can be found by imagining a right triangle in the cone, where one side is the height (h) of the cone and the other is the radius (r) of the circle. This makes s the hypotenuse of this triangle and can be found with the pythagorean theorem
s = √(h2 + r2)