## Monday, February 7, 2011

### Math on Television

I spent a couple hours on the floor this weekend, re-routing wires around the back of my television set and sound system. In the process of doing this, I rediscovered a few secrets about TV screen sizes.

(Go here to refresh your mind on measuring screens)

Our old TV was a Sony WEGA 32" tube-type model. Here's an old photo of it, in its cabinet. The screen is 32 inches diagonally. It has a 4:3 ratio of width to height. What are the screen's dimensions?

I forgot to measure it before I gave the TV away, but given this information, we can figure it out. That's what math is for, right?

Our latest TV (which we got from a friend) is a 40" Sony Bravia. That means 40 inches diagonally. It has a 16:9 ratio of width to height. It's much wider so I had to cut the top of my TV stand.

What are the screen's dimensions? I could measure the TV when I get home today, but that's no fun. Let's calculate its size too.

You learned in elementary school (using Excel Math, I hope!) that the length of the diagonal c (the hypotenuse) of a triangle is related to the other two sides of the triangle. The formula that describes their relationship is called the Pythagorean Theorem and in layman's terms is stated this way:

a2 + b2 = c2.

We need this formula for our calculations. Here's my work on calculating the dimensions of the first Sony TV:

It is 32 inches diagonally and it has an a/b ratio of 4:3. Diagonal c = 32 and c2 is 1024.

I now need to find numbers for a and b where two things are true at the same time:

a2 + b2 = 1024 and simultaneously 3a = 4b

If you look at my work on the whiteboard, you will see I came up with a = 25.6 and b = 19.2

I did this by trial and error - I just chose some dimensions and calculated until I found the right answers. Now on to the new set. We use the same process.

It is 40 inches in diagonally, and has an a/b ratio of 16:9. Diagonal c = 32 and c2 is 1600.

I now need to find numbers for a and b where two things are true at the same time.

a2 + b2 = 1600 and at the same time 9a = 16b

If you look at my work on the whiteboard, you will see I came up with a = 34.85 and b = 19.6

Let me explain in more detail. I did this by trial and error - I just chose some dimensions and calculated until I found the right answers. You can do it too, there's no fancy math here:

1. choose a number a , say 35 and multiply it by itself (square it) a= 35 x 35 = 1225
2. check to make sure that's smaller than c2 or 1600. It is.
3. multiply a by 9 to get 315, then divide 315 by 16 to get b = 19.7 (this is to ensure the dimensions match the 16:9 ratio of our set)
4. multiply 19.7 x 19.7 (square it) to learn that b2 =388
5. add 388 and 1225  and get 1613 - is this sum equal to 1600 or c2 ? No, it's too large.
6. start over with a smaller a, such as 34.9 and see what happens.

The numbers in red on each screen represent the square inches of screen area or a x b.

You can see the new TV is larger than the old one. But how much larger? How would you calculate the difference?

You might do it this way 40 inches divided by 32 inches = 1.25 times larger diagonally.

You might do it this way 683 divided by 492 = about 1.4 times larger in surface area.

Same TV sets, same math, almost-but-not-quite-the-same question = different answers.

(I'm home. I measured. The calculations were correct!)