Advertisers aren't exactly beacons of truth. Companies create ads to sell products, not to be honest - and these two goals aren't always aligned with one another. As a result, it's only natural to be skeptical when you see an ad like this:
An 80-inch screen is impressive, no doubt, but does it really have "more than double the screen area of a 55-inch TV"? After all, 80 isn't twice 55. At first glance, this may seem like self-serving hyperbole.
But let's not be so hasty. It's easy to overlook the fact that the ad is comparing the areas of the screens, not the lengths of the diagonals. Before we pass judgment on this commercial, we need to translate the information we have about the diagonals into information about the areas.
Now, just knowing the diagonal of a rectangle isn't enough to determine its area. For example, in the picture below, the red rectangle has the smallest area, followed by the blue, followed by the green, even though the diagonals of each shape are equal:
Thankfully, we also know something else about the Sharp television - its aspect ratio. Like most televisions sold nowadays, both the 55-inch and the 80-inch Sharp screens have an aspect ratio of 16:9. This means that for every nine units of height, there are sixteen units of width; equivalently, it means that the width of each screen is 16/9 times its height.
Since we don't know the height or the width of either TV, to compute the screen area we need to use our good friend algebra. For the 55-inch TV, let's call the height h. This means the width is 16h/9, and by the Pythagorean theorem, we know the sum of the squares of the lengths is equal to the square of the diagonal. But we know the length of the diagonal - it's 55 inches! Pulling this all together gives us the equality
h2 + (16h/9)2 = 552.
Both terms on the left hand side have a factor of h2, so adding them together simplifies the left hand side to approximately 4.16h2. After dividing each side by 4.16 and taking square roots, we obtain h ≈ 26.97 inches.
But once we know the height, we know the width, too! In this case, it's 16/9 x 26.97 ≈ 47.95 inches. And as with any other rectangle, once we know the width and the height, we can multiply them together to find the area, which here gives us approximately 1,293.21 square inches.
This method of combining the Pythagorean Theorem with information about the aspect ratio and the diagonal works equally well for the 80-inch screen - all we need to do is replace 55 with 80 in our first equality. I'll spare you the details, but in the second scenario we find that the area is approximately 2,734.42 square inches. The ratio of the two screen areas is then 2,734.42 ÷ 1,293.21, or around 2.11, a bit more than two. It looks like Sharp's ad team was telling us the truth after all!
There are plenty of extensions you could think about next. For example, can you come up with a general formula to find the area of a screen given the diagonal and the ratio between width and height? What size screen would Sharp need to build in order for them to advertise it as having three times the area of the 55-inch? Four times? If Sharp came up with a 152" screen to rival this one from Panasonic, how do you think they'd advertise it?
Beyond size, what do you think the next revolution in home entertainment technology will be? Whatever the case, I look forward to investigating the mathematics behind it.
Teachers: Interested in exploring the relationship between diagonals and screen areas with your students? Then check out our latest lesson, Viewmongus (the lesson formerly known as Pythagorean TV)!