Consider a cube with sides that are
exactly 10 centimeters long, as shown. What is the volume of the cube?
The volume is the space a shape
occupies. For a rectangular prism or a cube,
it’s equal to the length times the width times the height. The cube has sides that are all 10
centimeters long. So its volume is 1000 centimeters
cubed. I’m going to keep the volume on
screen so we can refer to it later in the problem.
What is the surface area of the
The surface area is the total area
on the surface of a shape. To find the surface area of our
cube, we could imagine unfolding it and then finding the total area of the resulting
squares. Our cube has six sides, each of
which is 10 centimeters by 10 centimeters. So the surface area is 600
centimeters squared. Again, I’m going to save this
number on screen so we can look at it in the next part of the problem.
If the cube were cut in two, would
the total volume and surface area increase, decrease, or stay the same?
So here’s the cube as if it were
cut in half vertically, which would give us two rectangular prisms with sides of
five centimeters, five centimeters, and 10 centimeters. Again, the volume of a rectangular
prism is the length times the width times the height. And we’ll want to multiply this by
two since we want the total volume of both of these rectangular prisms. And if we plug everything in, the
rectangular prisms have a total volume of 1000 cubic centimeters.
Now, let’s find the surface
area. We have two rectangular prisms with
two sides that have an area of 10 centimeters times 10 centimeters. And they each have four sides that
have an area of five centimeters times 10 centimeters. Altogether, that gives us a total
surface area of 800 centimeters squared.
Now, let’s compare the cube and the
rectangular prisms. The volume of both the cube and the
rectangular prisms is 1000 cubic centimeters, which makes sense, as we created the
rectangular prisms from the cube. So they should be able to occupy
the same space. While the volume was constant, the
surface area was not. It increased when we cut the cube
in two. So when we cut our cube or any
shape in general into more pieces that are smaller, the total volume would stay the
same. And the total surface area would