Because a Geek Has to Do it: Calculating the Dimensions of a New TV

1080p - the "p" stands for Pythagoras.
1080p – the “p” stands for Pythagoras.

The bulb in our 10 year-old DLP rear-projection TV burnt out for the second time in its lifespan earlier this week. The last time I had to replace it, my journey to secure a bulb ended in haggling with the rather unsavory proprietor of a strip-mall TV repair outlet. As a family, we decided it was time to put the old gray mare out to pasture, and give ourselves a little upgrade.

The question then became, how much TV could we fit in the same spot in our living room? The old TV, which we’d been quite happy with for so long, had a 49″ diagonal screen. But, being an old “narrow” rear-projection unit, the bezel around the screen was somehting like 7″, and the unit was about 16″ deep. Looking at the new sets, with nearly no bezels, and needing to know first how tall they’d be (to make sure we could fit the box in our car), and second how wide it’d be (to fit on the current shelf without obscuring a light switch, it was time to crank out a little old-fashioned math.

TV sizes are given in their diagonal dimension, making it easy to do the math if we know the Pythagorean Theorem: a² + b² = c². The diagonal dimension is the hypoteneuse of a right triangle, with the height and width being the other two sides. One other handy fact we know is that almost all HDTVs are built in a 16:9 aspect ratio; the ratio of the width to the height is 16:9. Another way to put this is that a/b = 16/9. We have everything we need to figure out the dimensions of the various available sets.

First, I was going to see if we could get a really big set: 70″ That means c=70.

If we solve the ratio equation for a, we get a=16b/9.

Plugging those into the theorem we get:
a² + b² = c²
(16b/9)² + b² = (70)²
(256b²/81) + b² = 4900
256b² + 81b² = (4900)(81)
337b² = (4900)(81)
b² = (4900)(81)/(337) = 1177.1
b = 34.3

So a 70″ diagonal TV set has a screen that’s 34.3″ tall. Knowing a = 16b/9, we can easily find that a = 61; the screen is 61″ wide (sans bezel). Such a screen would have been too wide for our location.

But, with the above math all set up, it was an easy process to replace (70)² with (65)² and (60)² to get get the range of values. QED!

[ps. we picked up a 60″ Vizio Smart TV on sale at Costco. It’s beautiful!]

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