A pyramid is on an equilateral triangular base of 21 centimeters and is 23 centimeters high. How long, to the nearest hundredth, is the pyramid’s lateral edge?
Let’s begin by sketching this pyramid. We know that there is an equilateral triangle on the base. So, if we label the base with 𝐴, 𝐵, and 𝐶, we know that all three sides will be 21 centimeters. We are also given that the pyramid is 23 centimeters high. And if we label the vertex with 𝐷, then we can actually put this measurement inside the pyramid. That’s because when we talk about the height of a pyramid, we mean the perpendicular height. And that’s the distance from the vertex 𝐷 to the centroid of the base.
We are asked in this question to find the lateral edge of the pyramid. And assuming that this is a right regular pyramid, then that’s any of the lengths from the vertex to the base, for example, this line in pink here, which is the line segment 𝐴𝐷. We can observe that we can create another line segment from 𝐴 to the centroid. And then we have created a right triangle within the pyramid.
Given that we wish to calculate the length of this lateral edge 𝐴𝐷, if we knew the length of the line from the vertex 𝐴 to the centroid, then we would have enough information to calculate the length of the line segment 𝐴𝐷. So, let’s see how we can calculate this base length from 𝐴 to the centroid. To do this, let’s take a closer look at the two-dimensional equilateral triangle at the base of the pyramid.
We know that all three sides will be 21 centimeters. We also know that the centroid of the triangle is formed at the intersection point of the three medians. So, let’s consider the median from 𝐴 to the centroid, and let’s define its length to be 𝑥 centimeters. Remember that this is the same length in the pyramid that we need to calculate. So, in order to calculate 𝑥, the first thing we need to do is calculate the whole length of the median from vertex 𝐴. Let’s define the length of this median to be 𝑦 centimeters. And we can recall that the median in an equilateral triangle is a perpendicular bisector.
So, we have a right triangle and the length from 𝐵 to the midpoint of 𝐵𝐶 will be 21 over two centimeters. As we have the length of two sides in a right triangle and one that we wish to find out, we could apply the Pythagorean theorem. This theorem states that the square of the hypotenuse is equal to the sum of the squares on the other two sides.
In this triangle then, the two shorter sides will be the length of the median, which is 𝑦 centimeters, and 21 over two centimeters. And the hypotenuse is 21 centimeters. Substituting these into the Pythagorean theorem then, we have 𝑦 squared plus 21 over two squared equals 21 squared. This simplifies to 𝑦 squared plus 441 over four equals 441. We can then subtract 441 over four from both sides, which leaves us with 𝑦 squared is equal to 1323 over four. We can then take the square root of both sides of this equation, which leaves us with the fact that the median is 21 root three over two centimeters.
Now that we have the length of the median, we need to work out what proportion of that median that 𝑥 is. And we can do that by recalling the centroid theorem. This tells us that the distance from each vertex to the centroid is two-thirds of the length of the median from this vertex. This means that in our triangular base, we can say that 𝑥 is equal to two-thirds of 𝑦. And since 𝑦 is equal to 21 root three over two, we have that 𝑥 is equal to two-thirds times 21 root three over two. We can take out a common factor of two and then take out a common factor of three, leaving us with 𝑥 is equal to seven root three.
And now we have the length from the vertex to the centroid. This is the same as we had in the pyramid that we needed to calculate in order to find the lateral edge length. We can then consider the two-dimensional triangle in the pyramid. And let’s define the lateral edge length to be 𝐿 centimeters. We observe that, once again, we can apply the Pythagorean theorem. This time, we have the two shorter side lengths of 23 centimeters and seven root three centimeters, which we can define as 𝑎 and 𝑏. The hypotenuse is 𝐿 centimeters.
Substituting these into the Pythagorean theorem, we have 23 squared plus seven root three squared equals 𝐿 squared. We can then evaluate the squares on the left-hand side as 529 and 147. And adding these values gives us 676. We then need to take the square root of both sides of this equation. So, we have that the square root of 676 is equal to 𝐿. 676 is a perfect square. So, we find that 𝐿 is equal to 26.
We have therefore determined that the pyramid’s lateral edge length is 26 centimeters. We can give the answer as 26.00 centimeters to indicate that this is to the nearest hundredth as required in the question.