Question Video: Finding the Angle between Two Face Diagonals of a Triangular Prism Mathematics

Video Transcript

The figure shows a right triangular prism. Find the angle between 𝐴𝐹 and 𝐴𝐶, giving your answer to two decimal places.

The first thing we can do in this question is identify our two lengths 𝐴𝐹 and 𝐴𝐶. The length 𝐴𝐹 will cut across the rectangular face here. The length 𝐴𝐶 will be the diagonal of the base of this right triangular prism. The angle between them will be the angle 𝐹𝐴𝐶 created here. And we can define this as the angle 𝜃. We can create a triangle 𝐴𝐹𝐶 in order to help us calculate our unknown angle 𝜃.

Let’s take a closer look at this triangle 𝐴𝐹𝐶. We know that the length 𝐹𝐶 is four centimeters and our angle here is 𝜃 at 𝐹𝐴𝐶. In order to apply trigonometry in this triangle, we need to be sure if we have a right triangle. We’re told in the question that this is a right triangular prism, which means that we have a right angle here at angle 𝐸𝐵𝐴 and at 𝐹𝐶𝐷. And since this plane on the bottom, 𝐵𝐴𝐷𝐶, meets our other plane 𝐸𝐵𝐶𝐹 at right angles, then we know that we have a right angle here at 𝐹𝐶𝐴.

We note, however, that as we look at our triangle 𝐴𝐹𝐶, we don’t quite have enough information. We’re going to need to find the length of one of these other two sides. If, for example, we look at 𝐴𝐶, we can see on our diagram that this length 𝐴𝐷, which is eight centimeters, is different to the length of 𝐴𝐶. We’ll need to create another right triangle in two dimensions to help us find the length of this side 𝐴𝐶.

We can, in fact, create this triangle 𝐴𝐶𝐷, which will have a right angle at angle 𝐴𝐷𝐶. We can draw out triangle 𝐴𝐶𝐷 on the bottom of our right triangular prism. We can see that 𝐶𝐷 is three centimeters and 𝐴𝐷 is eight centimeters. Don’t worry if your diagrams aren’t perfectly accurate; they don’t have to be to scale. They’re just there to help us visualize the problem. Remember why we’re doing this. We’re trying to find the length 𝐴𝐶, which is common to both triangles. We can define this as anything, but let’s call it the letter 𝑥.

When we find this length 𝑥 on our first triangle, we can fill in the information into our second triangle. As we have a right triangle and two known sides and one unknown side, we can apply the Pythagorean theorem, which tells us that the square of the hypotenuse is equal to the sum of the squares on the other two sides. So we take our Pythagorean theorem, often written as 𝑐 squared equals 𝑎 squared plus 𝑏 squared. The hypotenuse, 𝑐, is our length 𝑥. So we’ll have 𝑥 squared equals three squared plus eight squared. That’s the length of our two other sides. And it doesn’t matter which way round we write those.

We can evaluate three squared is nine, eight squared is 64, and adding those gives us 𝑥 squared equals 73. To find 𝑥, we take the square root of both sides of our equation. So we have 𝑥 equals the square root of 73. It’s very tempting at this point to pick up our calculator and find a decimal answer for the square root of 73. But as we haven’t finished with this value, we’re going to keep it in this square root form.

Now that we have found 𝑥, that means we’ve found our length of 𝐴𝐶. And so we can use this to find our angle 𝜃. As we’re interested in the angle here, that means we’re not going to use the Pythagorean theorem again, but we’ll need to use some trigonometry. In order to work out which of the sine, cosine, or tangent ratios we need, we’ll need to look at the sides that we have.

The length 𝐹𝐶 is opposite our angle 𝜃. 𝐴𝐶 is adjacent to the angle. And the hypotenuse is always the longest side. Now, we’re not given the hypotenuse, and we’re not interested in calculating it, so we can remove it from this problem. Using SOH CAH TOA can be useful to help us figure out which ratio we want. We have the opposite and the adjacent sides, so that means that we’re going to use the tan or tangent ratio. tan of 𝜃 is given by the opposite over the adjacent sides. We now fill in the values that we have.

The opposite length is four centimeters, and the adjacent length is given by root 73. So tan of 𝜃 is equal to four over root 73. In order to find 𝜃 by itself, we need the inverse operation to tan. And that’s finding the inverse tan. This function, written as tan with a superscript negative one, can usually be found on our calculator above the tan button. Using our calculator to evaluate this will give us 𝜃 equals 25.0873 and so on. And the units here will be degrees as, of course, this is an angle, not a length.

We’re asked to round our answer to two decimal places. So that means we check our third decimal digit to see if it’s five or more. And as it is, then our answer is given as 25.09 degrees. And so the angle between 𝐴𝐹 and 𝐴𝐶 is 25.09 degrees.

Before we finish with this question, let’s just review what we could have done instead. When we started this question, we had this large triangle 𝐴𝐹𝐶 which cut through our triangular prism. We were told that 𝐹𝐶 was four centimeters. And we worked out this length of 𝐴𝐶. But could we have done it by working out the length of 𝐴𝐹 instead?

If we look at our triangular prism, in order to work out the length of 𝐴𝐹, we’d need another triangle. We do have a right triangle here. And this length of 𝐸𝐹 will be eight centimeters. However, if we were trying to work out this length of 𝐴𝐹, we’d also need to work out this length of 𝐴𝐸. In order to find 𝐴𝐸, we’d need to create yet another right triangle. We would know that 𝐴𝐵 is four centimeters and 𝐴𝐵 is also three centimeters. So we would eventually get the correct answer for 𝐴𝐹 and, therefore, our angle 𝜃. It’s just that that second method would involve three triangles instead of the two triangles that we used.