WEBVTT
00:00:01.850 --> 00:00:12.380
Determine the base side length of a right square pyramid whose height is 12 centimeters and volume is 1,296 cubic centimeters.
00:00:12.880 --> 00:00:16.050
It can be helpful to draw a sketch of the information.
00:00:16.320 --> 00:00:18.020
So here we have a pyramid.
00:00:18.320 --> 00:00:22.870
And because it’s a square pyramid, that means that the base is a square.
00:00:22.980 --> 00:00:31.490
The fact that this is a right pyramid means that the apex, or the top of the pyramid, lies above the centroid of the base.
00:00:31.970 --> 00:00:37.900
The only length measurement information we’re given here is that the height is 12 centimeters.
00:00:38.270 --> 00:00:47.470
And so in order to work out the base side length, we’ll need to use the fact that the volume is 1,296 cubic centimeters.
00:00:47.790 --> 00:00:51.880
We can remember that there is a formula regarding the volume of a pyramid.
00:00:52.240 --> 00:01:01.560
This formula tells us that the volume is equal to one-third times the area of the base multiplied by ℎ, which is the height of the pyramid.
00:01:01.780 --> 00:01:06.540
In this question, we are given the volume and the height of the pyramid.
00:01:06.670 --> 00:01:11.750
What this formula would allow us to do then is simply work out the area of the base.
00:01:12.010 --> 00:01:19.300
However, if we did use this formula to work out the area of the base, that would allow us to work out the side length of the base.
00:01:19.500 --> 00:01:22.280
That’s because we know that the base is a square.
00:01:22.690 --> 00:01:26.770
Now we can apply this formula and fill in the values that we know.
00:01:26.960 --> 00:01:39.270
The volume is 1,296, and the height is 12, so we have 1,296 is equal to one-third times the area of the base times 12.
00:01:39.630 --> 00:01:45.310
On the right-hand side, we can simplify one-third times 12 as four.
00:01:45.620 --> 00:01:53.080
Dividing both sides by four, we have 1,296 over four is equal to the area of the base.
00:01:53.380 --> 00:02:01.480
Simplifying on the left-hand side, we have now worked out that the area of the base is 324 square centimeters.
00:02:01.880 --> 00:02:09.240
At the start of this question, we recognized that finding the area of the base would allow us to find the side length.
00:02:09.540 --> 00:02:12.860
This is because we know that the base is a square.
00:02:13.050 --> 00:02:20.650
To find the area of a square, if we say that the side length of the square is 𝑙, then the area is equal to 𝑙 squared.
00:02:21.010 --> 00:02:28.540
We have already worked out that the area of the base, that’s the area of the square, is 324 square centimeters.
00:02:28.810 --> 00:02:36.690
And so if we take the side length on this square to be 𝑙, then 𝑙 squared must be equal to 324.
00:02:37.070 --> 00:02:41.480
To find the value of 𝑙, we would take the square root of both sides.
00:02:41.980 --> 00:02:45.640
And the square root of 324 is 18.
00:02:45.870 --> 00:02:50.040
And we know that that’s going to be a positive value because it’s a length.
00:02:50.670 --> 00:02:57.090
So we can give the answer that the base side length of this square pyramid is 18 centimeters.