Let a, b, and c be the height, width, and the diagonal of the screen respectively
Since they form a triangle and the angle between a and b is 90 degrees, according to the Pythagorean Theorem a^2 + b^2 = c^2
Let's assume that the aspect ratio is 16:9. Therefore b/a = 16/9
Now we have a system of two equations with two unknowns. We need to express everything in terms of a single unknown, so let's express a in terms of b.
Since, b/a = 16/9, by cross multiplying 16a = 9b. Therefore a=9b/16.
Now we can subsitute 9b/16 for a in the first equation
a^2+b^2=c^2 becomes (9b/16)^2 + b^2 = c^2
Squaring the nominator and the denominator of the fraction yields
81b^2/256 + b^2 = c^2
To perform the addition, we need to express b^2 as a fraction with denominator 256 which is 256b^2/256, Now we have
81b^2/256 + 256b^2/256 = c^2
Adding fractions with the same denominators is now easy and produces
337b^2/256 = c^2
Cross multiply to get
337b^2 = 256c^2
Divide both sides by 337 to get b^2
b^2=256c^2/337
Take the square root of both sides to get b
b = sqrt(256c^2/337)
Lots of complicated Pythagorean responses. This is much simpler to answer with Trigonometry.
TLDR; just multiply the diagonal by the following two coefficients, and add the bezel width, to get the width and height:
w = diag x 0.872 + 2 x bw
h = diag x 0.49 + 2 x bw
(Where diag is the diagonal size of the screen and bw is the bezel width)
Thats all you need to know to do this.
Now here’s the Teig behind it:
First, calculate the angle of the diagonal relative to the base, using tangent, which yields ::
a = tan(9/16)
This will give you an angle of 0.512 radians (or roughly 30 degrees).
Next, calculate the ratio of the height relative to the hypotenuse (the diagonal length) using arc-sine, which yields the relative proportion of opposite length over the hypotenuse for a given angle:
h = asin(0.512)
The result will be roughly h = 0.49, or 49% of the diagonal.
Finally, get the ratio of eidth to hypotenuse using arc-cosine:
w = acos(0.512)
The result is roughly w = 0.872, or 87.2% of the diagonal.
that’s all the math! Enjoy!
