How to Find the Volume and Surface Area of a Cube and Rectangular Prism

A common task in geometry is to find the volume and the surface area of a cube and a rectangular prism. These questions are very easy if you memorize the associated formulas, but rather than memorize and possibly forget them, it is far better to simply understand them. It is also important to state your answer using the proper units, since not doing so will typically result in the question being marked partially or completely wrong. This article shows the steps for solving problems on this topic. It assumes that the reader has the necessary prerequisite knowledge.

Instructions

1. 

   First, it's essential to understand the difference between volume and surface area. In simplest terms, volume means the space inside an object. For example, we may want to know the volume of a cardboard box, meaning how much it can hold. We measure volume in cubic units (units³). This is because we compute volume by effectively multiplying three dimensions (such as length times width times depth).

2. 

   Now let's define surface area informally. Surface area would relate to how much paint we would need if we wanted to paint the entire exterior of an object, such as a brick, on all 6 faces. We're not concerned about the space inside of it like we were when talking about the volume inside a box. Here we are simply going to compute the area of each of the faces, and then add those areas together. We measure surface area in square units (units²). This is because surface area is just the sum of individual areas, and each area is computed by effectively multiplying two dimensions (such as length times width).

3. 

   Take a look at this rectangular prism. You can think of it as an empty cardboard box. Notice how its dimensions are labeled as length, width, and depth. It's important to understand that it doesn't matter what we call the dimensions. In fact, very often you'll see height in place of one of these three terms. All that matters is we're dealing with three perpendicular dimensions.

4. 

   A common question is to compute the volume of a rectangular prism. You probably remember that for a 2D (flat) rectangle, the area is computed as length times width. To find this volume of this 3D object, we'll just extend that formula to three dimensions, and multiply the dimensions shown, to get the formula V=length times width times depth, or V=LWD. We must always express our answer in units³. For example, if we were told that the three dimensions involved in this object were 3 ft., 4 ft., and 5 ft., we would multiply them together to get a volume of 60 ft.³ (sixty cubic feet).

5. 

   Now let's find the surface area of that box. Since we're not dealing with volume, don't think any more about the space inside the box, but instead think that we're going to be painting the six exterior surfaces (faces) of the box. To do this, we need to compute the area of each face. Each face is a simple 2D (flat) rectangle.

   You probably remember that the area for a rectangle is just length times width, or simply to multiply the two dimensions. Look at the yellow side of the box. It's area is L times W, or LW, omitting the operation sign. The orange side has an area of length times depth, or LD, using the dimension names given. The turquoise side has an area of width times depth, or WD.

6. 

   Make sure you see that each of these three faces is repeated on the other side, which we can't see. For example, the top of the box has the same area as the bottom, and so forth. This means that if we wanted to add up our areas, we would have to do LW+LW+LD+LD+WD+WD. We can combine like terms to get SA (surface area) = 2LW+2LD+2WD.

   Make sure the formula makes sense to you. Using the dimensions above of 3, 4, and 5 ft, our surface area formula would be SA=2(3)(4)+2(3)(5)+2(4)(5), which equals 94. We would express our answer is 94 ft² (ninety-four square feet) since we're dealing with area (albeit a sum of areas). Don't get mixed up and think that we are to square the number 94. We just square the units of feet to show that we're really dealing with feet times feet.

7. 

   Doing all this for a cube is much easier. Let's compute the volume of a cube. We know that we must multiply all three dimensions, but for a cube, all the dimensions are equal. Call each one an edge. Our volume formula is simply edge times edge times edge, which can be simplified as V=e³. If the cube had an edge of 4 in., its volume would be 64 in³.

8. 

   Finding the surface area of a cube is also easy. We know we have to add up the area of each face. Each face simply has an area of e times e, or e². There are six such faces, so our surface area is six times that, giving us SA=6e². You may see another variable used for the dimension of the cube such as s for side, but it's the same thing. For example, if a cube had an edge of 5 cm., the surface area would be SA=6(5)² or 150 cm².

9. 

   That's all there is to it. Rather than memorize the formulas, it is far better to understand where they come from, and practice using them. Remember to express volume in cubic units, and surface area in square units. ☺

Resources

• Math with Larry (Free Online Math Help)