Surface area, at its simplest and most basic, means (a) find the area of each face of a 3-D object, then (b) add
When your 3-D object is a prism, there are some formulas that can be applied and some steps to follow that can help to organize the different calculations needed.
These formulas are limited to orthogonal prisms (sometimes called "right" prisms) --- those in which the edges linking the base faces together are perpendicular to the base faces.
In other words, the edges of the "sides" are perpendicular to the "top" and "bottom". This creates side faces that are rectangles rather than parallelograms.
Step 1
Make room on your paper for at least three separate sub-problems. This process will not be done with a single formula.
Step 2
Determine which face is the base face. Keep in mind that the drawing may be angled so that the base face is not at the "bottom", but rather the front or the side. Prisms always have rectangular faces, so if any of the faces is NOT a rectangle, then that's your base face.
For our example, we will use this triangular prism, whose base face is the front:
Step 3
Make sure that you have the necessary measurements for that base face.
- If it's a triangle or a parallelogram, you will need its base and height.
Our example:
- A trapezoid will need both bases and the height.
- A hexagon or octagon or something like that will need the apothem and a side length.
- A rhombus will need both diagonals, or you can treat it like a parallelogram and find the base and the height. Either will work; pick the one that has nicer numbers.
Step 4
Find the area of the base face. Obviously, different shapes will need different formulas. For more information, consult the article How to Find Area.
Since our example has a triangular base, we multiply the base by the height, then cut it in half (which can be done by multiplying by 1/2 or can be done by dividing by 2).
Step 5
Double that area. There is a top and a bottom, and they are the same, so we multiply by two.
Step 6
Set that subtotal to one side. Label it "base area" so you can find it later.
Step 7
Find the perimeter of the base face.
Step 8
Set that subtotal to one side. Label it "perimeter", so you don't get it confused with the other subtotal.
Step 9
Identify the height of the prism. The height is the distance between the two base faces --- the top and the bottom. But keep in mind that if your drawing is on its side, so that the base is really the front, then your height will be the distance from front to back.
Step 10
Multiply the perimeter with the height. You have now found the surface area for all of the lateral sides (i.e. right and left, etc., but not top and bottom).
Step 11
Set that subtotal to one side. Label it "lateral area" so you can find it again.
Step 12
Take this last subtotal, the lateral area, and add it to the first subtotal, which was the base area. You are now done.
Tips
- Labeling your subtotals is extremely important!
- If all else fails, treat each face of the shape as a separate entity. Find the area of each one, then add them all up.
Warnings
- Do not try to make this process into one formula.
- Do not confuse "base face" with "base". The base face is the 2-D shape at the "bottom" of your prism. But that 2-D shape may also have a base, meaning a measurement along one edge that will be used when calculating its area.
- Do not confuse the "height" of the triangular 2-D base face, used when calculating the base face area, with the "height" of the prism itself, which is actually the distance between the orthogonal base faces (as shown above in figure under Step 9).
