Video Transcript
𝑀𝐴𝐵𝐶 is a regular pyramid whose base 𝐴𝐵𝐶 is an equilateral triangle whose side length is 32 centimeters. If the length of its lateral edge is 88 centimeters, find the height of the pyramid to the nearest hundredth.
Let’s begin with a sketch of this pyramid 𝑀𝐴𝐵𝐶. We are told that 𝐴𝐵𝐶 is an equilateral triangle, so all its sides will be of length 32 centimeters. We are also told that the lateral edge is 88 centimeters. So, for example, we could say that that means that the length of 𝑀𝐴 is 88 centimeters. The height of this pyramid is the perpendicular distance from the vertex 𝑀 to the centroid of the base. Let’s observe that we could create this right triangle within the pyramid. We want to calculate ℎ, the perpendicular height. We know that the lateral edge is 88 centimeters. So if we could determine this length from the vertex 𝐴 to the centroid of the base, then we would be able to calculate the value of ℎ. So, let’s consider how we can calculate this distance from 𝐴 to the centroid.
Let’s create a two-dimensional drawing of the base of the pyramid, which is the equilateral triangle 𝐴𝐵𝐶. Let’s define this distance from vertex 𝐴 to the centroid as 𝑥 centimeters in the pyramid. Well, on the base of the pyramid, it is here on this equilateral triangle. If we extended this pink line, we would have created the median of the equilateral triangle because the centroid of a triangle is formed at the intersection point of the three medians of the triangle. So, there are two things that we need to do. Firstly, we’ll need to work out the length of the median, and then we’ll need to work out the length of 𝑥.
The first property that we can use and recall is that the median of an equilateral triangle is a perpendicular bisector. So, that means that 𝐵𝐶 is split into two congruent pieces, and this median meets 𝐵𝐶 at right angles. So, given that we know the side length of this equilateral triangle, then we actually have enough information to apply the Pythagorean theorem in order to work out the length of the median. Since the side length of the equilateral triangle is 32 centimeters, then 𝐴𝐵 is 32 centimeters. The length from 𝐵 to the midpoint of 𝐵𝐶 must be half of 32 centimeters, so that’s 16 centimeters. We can define the length from 𝐴 to the midpoint of 𝐵𝐶 to be 𝑚 centimeters.
The Pythagorean theorem states that the square of the hypotenuse is equal to the sum of the squares on the other two sides. So, in this triangle, we have the two side lengths of 16 centimeters and 𝑚 centimeters, which can be 𝑎 and 𝑏, and the hypotenuse is 32 centimeters. Substituting these values into the Pythagorean theorem, we have 16 squared plus 𝑚 squared equals 32 squared. Evaluating the squares, we have 256 plus 𝑚 squared equals 1024. Rearranging this by subtracting 256, we have 𝑚 squared is equal to 768. Taking the square root of both sides, we have that 𝑚 is equal to the square root of 768. We can leave it as the square root of 768 or simplify it further to 16 root three. If we do convert this into a decimal, then we won’t round it just yet because we’ll need to use it in the next calculations.
So now we have calculated that the length of the median is 16 root 3 centimeters. We still need to work out this distance 𝑥, which is a proportion of the median. And in fact, it’s the centroid theorem which tells us this proportion. This theorem tells us that the distance from each vertex to the centroid is two-thirds of the length of the median from this vertex. What we have here then is that 𝑥 is two-thirds of 16 root three centimeters. We can then multiply two-thirds and 16 root three, and that would give us 32 root three over three centimeters. So finally, we have worked out that 𝑥 is 32 root three over three centimeters.
When we return to the pyramid, we can see that within this pyramid we now have this right triangle of which we know two lengths and we can calculate the height ℎ. We can clear some space for this calculation. It can be helpful to sketch this triangle from within the pyramid. So here we have the triangle with the hypotenuse of 88 centimeters, the base of 32 root three over three centimeters, and the side length of ℎ. Applying the Pythagorean theorem here, we would have ℎ squared plus 32 root three over three squared equals 88 squared. When we are squaring the values here, if we take this term 32 root three over three and square it, on the numerator, we’ll have 32 squared multiplied by root three squared, which is three, divided by three squared.
3072 over nine can in fact be simplified further to 1024 over three. On the right-hand side, 88 squared is 7744. We then rearrange by subtracting 1024 over three from both sides, leaving us with ℎ squared is equal to 22208 over three. We then take the square root of both sides, giving us ℎ is equal to the square root of 22208 over three. At this point, we then check to see how the answer should be given. And as we’re asked for the answer to the nearest hundredth, then we need to find a decimal approximation. This will be 86.0387 and so on centimeters. Rounding this to the nearest hundredth, we have the answer that the height of this pyramid is 86.04 centimeters to the nearest hundredth.
