Video: Calculating the Surface Area of a Regular Shape

Consider a cube with sides that are exactly 10 cm long: a) What is the volume of the cube? b) What is the surface area of the cube? c) If the cube were cut in two, would the total volume and the total surface area increase, decrease, or stay the same?

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Video Transcript

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 cube?

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 increase.

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