Pythagoras would have liked TV shopping

Pythagoras’ theorem is one of the beauties of mathematics. Something so simple as being able to work out sides of a right angled triangle has had pretty much unlimited use in practical engineering as well as very theoretical linear algebra and calculus.

For those of you less mathematically inclined, I’ll explain briefly what the theorem says and then show you a nice real-world application of it.

The Theorem

If your years of mathematics are long gone and forgotten, then Pythagoras’ theorem essentially says that:

Given a right angled triangle, the square of the length of the hypotenuse (longest side) is equal to the sum of the squares of the other two sides.

In other words given the triangle below, a2 + b2 = c2

What’s this about TVs then?

Suppose you wanted to buy a shiny new LCD TV for your living room. But you don’t know what screen size will look right in your room, or fit on your table. Now, suppose that the TVs you’re looking at don’t include the dimensions on the website. This is a hypothetical situation, but it can happen. All you know is that it is widescreen and you know the screen size in inches. Of course, the screen size they give you is the diagonal length of the screen. Crucially, for practical purposes, this doesn’t tell you how wide and high the viewing part of the screen will be (not including the framing around the edges).

So you see a 40” TV and wonder if it would be a reasonable size. Let’s see what information we have so far:

Ignoring the bit around the outside, we know the red length is 40 inches. It looks like we don’t have enough information to get the width and height. But, remember, this TV is widescreen. That means its aspect ratio is 16:9. In other words, if a screen was exactly 9 inches high, then it would be exactly 16 inches wide:

Now we’re getting somewhere. We can use Pythagoras’ theorem to work out the diagonal distance in red.

d2 = 162 + 92 = 256 + 81 = 337

Therefore, d is the square root of 337, which is 18.36 to 2 decimal places.

Now it’s easy, because we know all the proportions of our ‘typical’ widescreen TV. All we need to do is scale the sides so that they match our hypotenuse of 40 inches.

To get the scale factor, we must find out how much bigger our 40 inch TV is than the ‘typical’ TV screen we worked out:

scale factor = 40/18.36 = 2.1786…

Great! Now all we need to do is multiply our sides by the scale factor…

9 x 2.1786 = 19.61
16 x 2.1786 = 34.86

…and we’re done! Your 40 inch screen will be 34.86 inches wide and 19.61 inches high. Who needs dimensions on the packaging?! All hail the mighty Pythagoras!

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