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

The figure shows a right triangular prism. Find the angle between ๐ด๐น and ๐ด๐ถ, giving your answer to two decimal places.

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Video Transcript

The figure shows a right triangular prism. Find the angle between ๐ด๐น and ๐ด๐ถ, giving your answer to two decimal places.

The first thing we can do in this question is identify our two lengths ๐ด๐น and ๐ด๐ถ. The length ๐ด๐น will cut across the rectangular face here. The length ๐ด๐ถ will be the diagonal of the base of this right triangular prism. The angle between them will be the angle ๐น๐ด๐ถ created here. And we can define this as the angle ๐œƒ. We can create a triangle ๐ด๐น๐ถ in order to help us calculate our unknown angle ๐œƒ.

Letโ€™s take a closer look at this triangle ๐ด๐น๐ถ. We know that the length ๐น๐ถ is four centimeters and our angle here is ๐œƒ at ๐น๐ด๐ถ. In order to apply trigonometry in this triangle, we need to be sure if we have a right triangle. Weโ€™re told in the question that this is a right triangular prism, which means that we have a right angle here at angle ๐ธ๐ต๐ด and at ๐น๐ถ๐ท. And since this plane on the bottom, ๐ต๐ด๐ท๐ถ, meets our other plane ๐ธ๐ต๐ถ๐น at right angles, then we know that we have a right angle here at ๐น๐ถ๐ด.

We note, however, that as we look at our triangle ๐ด๐น๐ถ, we donโ€™t quite have enough information. Weโ€™re going to need to find the length of one of these other two sides. If, for example, we look at ๐ด๐ถ, we can see on our diagram that this length ๐ด๐ท, which is eight centimeters, is different to the length of ๐ด๐ถ. Weโ€™ll need to create another right triangle in two dimensions to help us find the length of this side ๐ด๐ถ.

We can, in fact, create this triangle ๐ด๐ถ๐ท, which will have a right angle at angle ๐ด๐ท๐ถ. We can draw out triangle ๐ด๐ถ๐ท on the bottom of our right triangular prism. We can see that ๐ถ๐ท is three centimeters and ๐ด๐ท is eight centimeters. Donโ€™t worry if your diagrams arenโ€™t perfectly accurate; they donโ€™t have to be to scale. Theyโ€™re just there to help us visualize the problem. Remember why weโ€™re doing this. Weโ€™re trying to find the length ๐ด๐ถ, which is common to both triangles. We can define this as anything, but letโ€™s call it the letter ๐‘ฅ.

When we find this length ๐‘ฅ 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 ๐‘ squared equals ๐‘Ž squared plus ๐‘ squared. The hypotenuse, ๐‘, is our length ๐‘ฅ. So weโ€™ll have ๐‘ฅ squared equals three squared plus eight squared. Thatโ€™s the length of our two other sides. And it doesnโ€™t matter which way round we write those.

We can evaluate three squared is nine, eight squared is 64, and adding those gives us ๐‘ฅ squared equals 73. To find ๐‘ฅ, we take the square root of both sides of our equation. So we have ๐‘ฅ equals the square root of 73. Itโ€™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 havenโ€™t finished with this value, weโ€™re going to keep it in this square root form.

Now that we have found ๐‘ฅ, that means weโ€™ve found our length of ๐ด๐ถ. And so we can use this to find our angle ๐œƒ. As weโ€™re interested in the angle here, that means weโ€™re not going to use the Pythagorean theorem again, but weโ€™ll need to use some trigonometry. In order to work out which of the sine, cosine, or tangent ratios we need, weโ€™ll need to look at the sides that we have.

The length ๐น๐ถ is opposite our angle ๐œƒ. ๐ด๐ถ is adjacent to the angle. And the hypotenuse is always the longest side. Now, weโ€™re not given the hypotenuse, and weโ€™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 weโ€™re going to use the tan or tangent ratio. tan of ๐œƒ 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 ๐œƒ is equal to four over root 73. In order to find ๐œƒ by itself, we need the inverse operation to tan. And thatโ€™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 ๐œƒ equals 25.0873 and so on. And the units here will be degrees as, of course, this is an angle, not a length.

Weโ€™re asked to round our answer to two decimal places. So that means we check our third decimal digit to see if itโ€™s five or more. And as it is, then our answer is given as 25.09 degrees. And so the angle between ๐ด๐น and ๐ด๐ถ is 25.09 degrees.

Before we finish with this question, letโ€™s just review what we could have done instead. When we started this question, we had this large triangle ๐ด๐น๐ถ which cut through our triangular prism. We were told that ๐น๐ถ was four centimeters. And we worked out this length of ๐ด๐ถ. But could we have done it by working out the length of ๐ด๐น instead?

If we look at our triangular prism, in order to work out the length of ๐ด๐น, weโ€™d need another triangle. We do have a right triangle here. And this length of ๐ธ๐น will be eight centimeters. However, if we were trying to work out this length of ๐ด๐น, weโ€™d also need to work out this length of ๐ด๐ธ. In order to find ๐ด๐ธ, weโ€™d need to create yet another right triangle. We would know that ๐ด๐ต is four centimeters and ๐ด๐ต is also three centimeters. So we would eventually get the correct answer for ๐ด๐น and, therefore, our angle ๐œƒ. Itโ€™s just that that second method would involve three triangles instead of the two triangles that we used.

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