Question Video: Finding the Length of the Diagonal of a Cuboid | Nagwa Question Video: Finding the Length of the Diagonal of a Cuboid | Nagwa

Question Video: Finding the Length of the Diagonal of a Cuboid Mathematics

A cuboid is shown in the figure. Find the angle between the diagonals 𝐴𝐺 and 𝐸𝐺. Give your answer to two decimal places.

05:15

Video Transcript

A cuboid is shown in the figure. Find the angle between the diagonals 𝐴𝐺 and 𝐸𝐺. Give your answer to two decimal places.

In this question, we’re asked to find this angle at 𝐴𝐺𝐸. It’s formed from this space diagonal 𝐴𝐺 and the face diagonal 𝐸𝐺. If we also highlight this line 𝐴𝐸, we can see that we have a right triangle formed.

Let’s take a closer look at this triangle 𝐴𝐸𝐺. We know that the length of 𝐴𝐸 will be 10 as that’s the same as the length of 𝐵𝐹. This angle 𝐴𝐺𝐸 is the one that we’re asked to find. We can use some trigonometry to find this angle. But we’d need to know some other pieces of information. We’d either need the length of 𝐴𝐺 or the length of 𝐸𝐺. Note that the length of 𝐸𝐺 will be different from this length of 30, which is 𝐻𝐺, as we have a diagonal. So that will be longer than the length of this side.

Let’s see if we can find a way to work out the length of 𝐸𝐺. If we look at the diagram of the cuboid, we can observe that this length of 𝐸𝐺 is also part of the triangle formed by 𝐸𝐺𝐻. This triangle is also a right triangle as we know that this forms the base of a cuboid, of which all of the faces would be rectangular. We can then draw out this triangle of 𝐸𝐺𝐻. Don’t worry if the diagram isn’t perfect. It doesn’t have to be to scale. It’s just to help us visualize the problem.

We can fill in any length information that we’re given. For example, 𝐸𝐻 will be the same as 𝐹𝐺. And that will be 15 units long. The length of 𝐻𝐺 is given as 30 units. We have this unknown length of 𝐸𝐺, which is the same as the one we’re finding for the triangle 𝐴𝐸𝐺. Let’s define this length as 𝑥. And then once we know that, we can use it in our second triangle.

When we have a right triangle and we have two lengths that we know and one that we want to find out, we can use the Pythagorean theorem, which tells us that the square on the hypotenuse is equal to the sum of the squares on the other two sides. We can then take the Pythagorean theorem and plug in the values that we know. The hypotenuse is the length 𝑥, and the other two sides are 15 and 30 units long. And it doesn’t matter which way we write those in the Pythagorean theorem.

So we’ll have 𝑥 squared equals 15 squared plus 30 squared. We evaluate the squares. 15 squared is 225, and 30 squared is 900. Adding those together gives us that 𝑥 squared is 1125. In order to find 𝑥, we take the square root of both sides. So we’ll have 𝑥 is equal to the square root of 1125. It might be tempting at this point to pick up our calculator and find this value as a decimal. However, we’ll leave it in the square root form as we’ll use it in the next calculation.

As we’ve found a value of 𝑥, that means we’ve found the length of 𝐸𝐺. It’s the square root of 1125. We can now use this to find our unknown angle. We can define this angle at 𝐴𝐺𝐸 to be 𝜃. So how would we go about finding the value of this angle?

As we have angles involved, the Pythagorean theorem won’t be very useful. So in this case, we’d have to apply some trigonometry. Looking at this triangle of 𝐴𝐸𝐺, we have the side 𝐴𝐸 that’s opposite our angle 𝜃. We also have the side that’s adjacent to it. That’s the side 𝐸𝐺. Note that the side 𝐴𝐺 is the hypotenuse of this right triangle. But as we’re not given it and we don’t want to find it, we can exclude it from our problem.

If we use the trigonometry phrase SOHCAHTOA, we observe that we have the opposite and adjacent sides. So that means we’re going to use our tangent or tan ratio. This ratio tells us that tan of the angle 𝜃 is equal to the opposite over the adjacent sides. We can then plug in the values that we have. So we have tan 𝜃 equals 10 — that’s the opposite side — over the square root of 1125, which was the adjacent side.

In order to find 𝜃 by itself, we need to use the inverse tan function. When we put this value on the right-hand side of our equation into our calculator, we might need to remember that this inverse tan function is often found above the tan button. We’ll often need to use the shift or second function key in order to be able to use the inverse tan function.

Using our calculator then, we get the value that 𝜃 is equal to 16.6015 and so on. And the units here will be in degrees because this is an angle and not a length. Rounding the answer to two decimal places means that we check our third decimal digit to see if it’s five or more. And as it isn’t, then our answer stays as 16.60 degrees.

So we’ve found that this angle 𝐴𝐺𝐸 was 16.60 degrees. And that was the angle that we were asked for between the diagonals 𝐴𝐺 and 𝐸𝐺.

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