WEBVTT
00:00:02.370 --> 00:00:06.640
Consider a cube with sides that are exactly 10 centimeters long, as shown.
00:00:07.070 --> 00:00:08.650
What is the volume of the cube?
00:00:10.300 --> 00:00:13.230
The volume is the space a shape occupies.
00:00:13.260 --> 00:00:18.200
For a rectangular prism or a cube, it’s equal to the length times the width times the height.
00:00:19.410 --> 00:00:22.260
The cube has sides that are all 10 centimeters long.
00:00:23.570 --> 00:00:26.660
So its volume is 1000 centimeters cubed.
00:00:28.420 --> 00:00:32.460
I’m going to keep the volume on screen so we can refer to it later in the problem.
00:00:34.390 --> 00:00:36.370
What is the surface area of the cube?
00:00:37.520 --> 00:00:41.140
The surface area is the total area on the surface of a shape.
00:00:41.790 --> 00:00:48.840
To find the surface area of our cube, we could imagine unfolding it and then finding the total area of the resulting squares.
00:00:50.140 --> 00:00:55.020
Our cube has six sides, each of which is 10 centimeters by 10 centimeters.
00:00:55.480 --> 00:00:58.700
So the surface area is 600 centimeters squared.
00:01:00.050 --> 00:01:04.330
Again, I’m going to save this number on screen so we can look at it in the next part of the problem.
00:01:06.260 --> 00:01:12.390
If the cube were cut in two, would the total volume and surface area increase, decrease, or stay the same?
00:01:13.390 --> 00:01:22.200
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.
00:01:23.320 --> 00:01:27.710
Again, the volume of a rectangular prism is the length times the width times the height.
00:01:28.400 --> 00:01:33.960
And we’ll want to multiply this by two since we want the total volume of both of these rectangular prisms.
00:01:35.280 --> 00:01:41.760
And if we plug everything in, the rectangular prisms have a total volume of 1000 cubic centimeters.
00:01:43.020 --> 00:01:44.560
Now, let’s find the surface area.
00:01:45.150 --> 00:01:52.600
We have two rectangular prisms with two sides that have an area of 10 centimeters times 10 centimeters.
00:01:53.150 --> 00:01:58.230
And they each have four sides that have an area of five centimeters times 10 centimeters.
00:01:59.560 --> 00:02:03.800
Altogether, that gives us a total surface area of 800 centimeters squared.
00:02:04.730 --> 00:02:07.940
Now, let’s compare the cube and the rectangular prisms.
00:02:08.360 --> 00:02:17.640
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.
00:02:17.860 --> 00:02:20.310
So they should be able to occupy the same space.
00:02:21.250 --> 00:02:24.620
While the volume was constant, the surface area was not.
00:02:24.740 --> 00:02:27.480
It increased when we cut the cube in two.
00:02:28.720 --> 00:02:36.190
So when we cut our cube or any shape in general into more pieces that are smaller, the total volume would stay the same.
00:02:36.280 --> 00:02:38.550
And the total surface area would increase.