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Calculating Diagonals

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Today we're going to talk about calculating diagonals. A diagonal is just a line that connects opposite corners of either a square or a rectangle. We'll look at how diagonals are related to right triangles, we'll talk about something called the Pythagorean theorem, and then we'll do some examples.

So let's look at how the diagonal of a square or a rectangle is related to a right triangle. So as I said before, the diagonal connects opposite corners of a square or a rectangle. So we can see that this diagonal is splitting this rectangle into two congruent or two same size triangles. And since we know that a rectangle has four right angles, or four 90 degree angles, we also know that these are going to be right triangles.

So let's think about how the sides of the rectangle relate to the sides of our right triangles. So first, we know that the horizontal and the vertical sides of our rectangle are at the legs of our right triangle. And then we can see that our diagonal is going to be the hypotenuse of our right triangles.

So let's go over the Pythagorean theorem and then we'll do an example using the Pythagorean theorem to calculate the diagonal of a rectangle. So my Pythagorean theorem says that for a right triangle with legs a and b and hypotenuse c, I can relate those three sides of the triangle with a squared plus b squared is equal to c squared. And the important thing to notice here is that a and b as the legs of our right triangle can be interchangeable. It doesn't matter which one is a or b. However, your c variable always needs to be the hypotenuse of your triangle.

So let's do an example. If I've got a rectangle with side lengths of 4.5 centimeters and 6.2 centimeters, I can see that if I want to calculate the diagonal this is going to form a right triangle. So I want to find what c is, which would be the hypotenuse of my right triangle or the diagonal of my rectangle when my side lengths are 4.5 centimeters or a and 6.2 centimeters or b.

So using my formula, I can see that c squared is going to be equal to 4.5 squared plus 6.2 squared. Simplifying this, I've got that 4.5 squared is going to give me 20.25 plus 6.2 squared is 38.44. And that's going to be equal to c squared. I can add these together. And so c squared will be 58.69.

Now to get c by itself, I need to cancel out the 2 exponent by taking the square root of both sides. And when you take the square root, you can actually get a positive and a negative answer. Since we're talking about the distance or side length, we know that we're only going to be needing to consider our positive answer. So I can say that c is going to be equal to the positive square root of 58.69, which is actually approximately 7.66. So my diagonal of my rectangle is going to be approximately 7.66 and that will be in centimeters.

So let's do another example using the Pythagorean theorem. We're using the Pythagorean theorem to find the hypotenuse of a right triangle, which, again, corresponds to the diagonal of a rectangle or a square. And so if we're using the formula to find the hypotenuse of this right triangle, we might want to have our formula in terms of just c instead of c squared.

So to do that I want to cancel out this 2 exponent. And I'll do that by taking the square root of both sides, which will cancel out this 2 exponent and I'll just be left with my formula in terms of c which is what I was looking for. And now on the other side I see that c is equal to the square root of a squared plus b squared. So let's use this version of the Pythagorean theorem to solve this next example.

So I've got a rectangle with side lengths of 3.7 inches and 1.9 inches and I want to find the diagonal, which, again, corresponds to the hypotenuse of my right triangle. So I'll call this c and I'm going to call my 1.9 inch side a and my 3.7 inch side b. So substituting those values into my formula, I have c is equal to the square root of 1.9 squared plus 3.7 squared. Simplifying underneath my radical and starting by squaring both of my values, 1.9 squared is going to give me 3.61. 3.7 squared gives me 13.69.

Adding these values together, I get 17.3. So this becomes the square root of 17.3. And now when you take the square root of a number, you can get a negative and a positive answer. But because we're talking about the diagonal of a rectangle, which is a distance, when we're using this formula we're only looking at the positive answer. So the square root of 17.3 gives me approximately 4.16. The diagonal of my rectangle is equal to 4.16. And that's in inches.

So let's look at our key points from today. Make sure that you get these into your notes if you don't have them already so that you can refer to them later. So we first talked about the fact that the diagonal of a rectangle or square corresponds to the hypotenuse of a right triangle. So then we can use the Pythagorean theorem, which we use for right triangles, to calculate the diagonal of a rectangle or square.

And then we looked at two different forms of the Pythagorean theorem. So you have to remember that c in our formula has to be the hypotenuse, but, again, a and b can be either of the legs of our right triangle. And the two forms of our formula were a squared plus b squared equals c squared. Or equivalently, just c is equal to the square root of a squared plus b squared.

So I hope that these key points and the examples that we did helped you understand a little bit more about calculating diagonals. Keep using your notes and keep on practicing and soon you'll be a pro. Thanks for watching.