Understanding Surface Area and Volume

Mentor: So, if we know that we can find the area of a two-dimensional figure,do you think that it is possible to find the area of a three dimensional figure? In fact, who can tell me what a three dimensional figure is?

Student: A three dimensional figure is like a ball or a cube--it's not flat.

Mentor: That's right. Now, can anyone say something about what it might mean to find the area of such a figure?

Student: When you say "find the area", do you mean the outside, or are the insides included?

Mentor: Well, it depends. There's actually no such thing as finding the "area" of a cube. Instead, we have the terms "volume" and "surface area". Let's talk about volume first. When you say "find the area" of a square, do you mean the outside, or are the insides included?

Student: We just look at the space it takes up on the paper; we assume the edges of the square have a width of zero.

Mentor: Precisely! Now imagine looking at the amount of space a cube takes up in 3 dimensions. We call that measure the volume of the cube.

Student: How do you measure the volume?

Mentor: Just as you measure and multiply the length and width of a rectangle to find its area, you multiply the length, width, and height of a 3-D object like a cube to find its volume. The three variables multiplied give it three dimensions, thus the volume, rather than simply the area. What do you suppose are the units for volume?

Student: Well, if there are three terms, all in inches, then it would be inches * inches * inches, which is inches cubed.

Mentor: How is this different from the units when you find area?

Student: Well, area is "squared" because you're just multiplying inches * inches.

Mentor: Exactly! Area is "squared" and volume is "cubed". How do you think that relates to their meaning?

Student: You find the area of a square or other two-dimensional objects, but you find the volume of three-dimensional objects like cubes!

Mentor: Good, we now know how to measure how much space an object takes up. But what about the outside of the object, as you mentioned earlier? What do you think "surface area" is?

Student: That sounds like it would be just the outsides--the area that is on the surface, which I can touch.

Mentor: Very well said! Surface area is the area of the surface of the three-dimensional shape. How would you calculate something like that?

Student: It seems almost too simple, but couldn't I just find the area of each two-dimensional face , then add the areas up?

Mentor: Absolutely! It is that simple. Almost all three dimensional objects you'll deal with are made up of two-dimensional faces that are just squares, triangles, etc, and the ones that are curved like spheres will have their own special formulas for surface area. Of course, the units for that are easy to find, right?

Student: Yep, it's just the standard units for area - units * units or units squared.

Mentor: You've got it! Now you're ready to try to solve some problems involving surface area and volume.