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We’re gonna need Pythagoras’ help to compare screen sizes in 2017

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Photo by James Bareham / The Verge

Android Central‘s Alex Dobie made this point to me in the buildup to Samsung’s Galaxy S8 launch yesterday: a 5.8-inch phone with the S8’s elongated 18.5:9 aspect ratio doesn’t have the same size screen as a 5.8-inch phone with the traditional 16:9. The two might share the same diagonal measurement, but in terms of area, the S8’s screen would be smaller. That’s because the change in aspect ratio breaks the linear scale by which we’ve compared almost all smartphones to date. If every rectangle has the same aspect ratio — the relationship between its height and width — then knowing its diagonal measurement gives us a rough way to compare or at least rank those rectangles by size.

Samsung and LG have broken from the industry convention with their new phones this spring — with the Galaxy S8 and the G6, respectively — and their renegade actions are wreaking havoc with our casual shorthand for comparing display dimensions. But never fear, there is still a way to bring them back into line and do direct comparisons. We’ll just need a little bit of Pythagorean mathematics to help us.

If you know the length of a right-angle triangle’s hypotenuse (c) and the ratio between its sides (a and b), you can work out the lengths of those sides and, consequently, the area of the rectangle within which that triangle resides.

Here’s my method of reverse-engineering a phone’s screen size from its diagonal measurement:

Convert from imperial to metric measurements, because everything’s simpler in metric form. For my example, let’s take the 5-inch / 127mm Google Pixel, which I could measure with a ruler to confirm my calculations were sensible.

Use Pythagoras’ theorem (a² + b² = c²) to figure out the three-way relationship between the diagonal line, which is the number we know, and the two straight sides. We already know the height and width are at a 16:9 ratio to one another, we just need to find out how the diagonal measurement relates to them.

16 x 16 + 9 x 9 = 337 = c²

√337 = c

c = 18.36

Next, divide the screen’s diagonal measurement (127mm) by our calculated value for c, and then multiply the result by our known values for a and b.

127 / 18.36 = 6.92

16 x 6.92 = 110.72

9 x 6.92 = 62.28

So now we know that the 5-inch Google Pixel has sides measuring 110.72mm in height and 62.28mm in width. (I pulled out a trusty ruler and verified this conclusion.) To know the area of the Pixel’s screen, we just multiply the two, and get a result of 68.95cm².

110.72 x 62.28 = 6,895.62mm²

Samsung Galaxy S8
Samsung Galaxy S8.
Photo by Amelia Holowaty Krales / The Verge

Now let’s do this all over again, this time with a theoretical 5.8-inch (147mm) 16:9 phone and the 5.8-inch 18.5:9 Samsung Galaxy S8.

First, the 16:9 phone, reusing the 18.36 value for c that we calculated above:

147mm / 18.36 = 8

16 x 8 = 128

9 x 8 = 72

128 x 72 = 9,216mm²

And now the Galaxy S8, with its own c calculation:

18.5 x 18.5 + 9 x 9 = 423.25 = c²

√423.25 = c

c = 20.57

147mm / 20.57 = 7.15

18.5 x 7.15 = 132.28

9 x 7.15 = 64.35

132.28 x 64.35 = 8,512.22mm²

So yes, at 85.12cm², the Galaxy S8 has a significantly smaller display area than a 16:9 phone with the same diagonal screen measurement (which would be 92.16cm²). In fact, it’s not a huge distance away from the Pixel’s 5-inch-diagonal 16:9 display, which is exactly how it feels in person. These measuring differences are more than academic: the nominally 5.8-inch Galaxy S8 is ergonomically similar to the Pixel, and the same is true of the 6.2-inch S8 Plus vis-a-vis the 5.5-inch Pixel XL. This would suggest that Samsung’s cramming a lot more screen into the same physical footprint, but in reality it’s only a moderate amount of extra screen.

Graph showing the scale of the Samsung Galaxy S8 (in the middle) relative to the Google Pixel and a theoretical 16:9 5.8-inch phone.
To scale: the Samsung Galaxy S8 (middle) compared to the Google Pixel and a theoretical 16:9 5.8-inch phone.
Diagram by Chaim Gartenberg / The Verge

(Additional complication: Samsung discounts the curved screen sides from its calculations of diagonal screen size, but does include the curved corners, so absolute precision is basically out the window with this class of display.)

The problem we now face, and which won’t be solved by this excursion into amateur mathematics, is how the hell we’re supposed to work out these differences at a glance. All I can say for certain now is that we should all start treating screen sizes on spec sheets with a lot more circumspection. Phone sizes are already highly variable owing to the differences in screen bezels, but now the displays themselves cannot be reliably compared either.