Two screens, the same diagonal, very different areas

Screens are sold by their diagonal. It is one number, it is easy to print, and it is a genuinely poor description of how much surface you are getting — because a diagonal fixes a length, and area needs a shape as well.

Getting the area from a diagonal and a ratio

Take a rectangle with an aspect ratio of w:h and a diagonal of D. Its sides are some common multiple of w and h, so call them kw and kh. Pythagoras gives D² = (kw)² + (kh)², so k = D / √(w² + h²). Multiply the sides together and the area falls out:

• Width = D × w / √(w² + h²)
• Height = D × h / √(w² + h²)
• Area = D² × wh / (w² + h²)

Everything about the shape is in that last fraction, wh / (w² + h²). The diagonal only sets the scale; the fraction decides how much of the possible area that scale actually buys.

The comparison worth remembering

Fix the diagonal at 27 inches and change nothing else.

For 4:3, the fraction is 12 / 25 = 0.48, and the numbers come out clean because √(16 + 9) = 5. The sides are 27 × 4 / 5 = 21.6 and 27 × 3 / 5 = 16.2 inches, so the area is exactly 21.6 × 16.2 = 349.92 square inches.

For 16:9, the fraction is 144 / 337 ≈ 0.427299703. The sides are 23.532540 and 13.237053 inches (both rounded), and the area is 104976 / 337 ≈ 311.501484 square inches.

Same diagonal. The 4:3 rectangle has 349.92 − 311.501484 = 38.418516 more square inches, and the ratio between them is exact: 0.48 × 337 / 144 = 337 / 300 ≈ 1.123333. A 4:3 rectangle has about 12.3 percent more area than a 16:9 rectangle on the same diagonal. Nothing in the diagonal hints at that.

Push it further with 21:9, which reduces to 7:3. The fraction is 21 / 58 ≈ 0.362069, the sides are 24.816916 by 10.635821 inches (rounded), and the area is 15309 / 58 ≈ 263.948276 square inches. Against that, the 4:3 rectangle is 349.92 / 263.948276 ≈ 1.325714 times as large — about 32.6 percent more area for an identical diagonal — and even the 16:9 is ahead by 311.501484 / 263.948276 ≈ 1.180161.

Why it goes that way

The fraction wh / (w² + h²) is largest when w = h, where it equals 0.5. A square is the shape that gets the most area out of a given diagonal, and every step away from square costs you some of it: 0.48 at 4:3, 0.427299703 at 16:9, 0.362069 at 21:9. Stretching a rectangle while holding its diagonal fixed always loses area, and the loss accelerates as the shape gets longer.

So a diagonal is only comparable between two things of the same aspect ratio. Across different ratios it is not a size at all — it is one measurement of a shape, quoted without the shape.

What the tool actually does

The free Aspect Ratio calculator takes three numbers — a width, a height and a new width — and returns two things: the ratio reduced to lowest terms, and the height that keeps that ratio at the new width. With its defaults of 1920 by 1080 and a new width of 1280, it prints 16:9 and 720.

The reduction is a greatest-common-divisor step on whole numbers, so give it whole pixel counts. The new height is the plain proportion new width × height / width, computed on the values as entered.

Two useful things follow. First, not every panel reduces to something friendly: 1366 by 768 has a greatest common divisor of only 2, so it reduces to 683:384 ≈ 1.778646, which is near 16/9 ≈ 1.777778 but is not equal to it.

Second: the arithmetic rounds badly at small sizes

A proportion returns whatever real number it returns, and pixels are whole. Resize a 1920 by 1080 image to a width of 150 and the exact height is 150 × 1080 / 1920 = 84.375. There is no 84.375-pixel-tall image, so you round.

• Round down to 84: the real ratio becomes 150 / 84 ≈ 1.785714, which is 0.4464 percent wider than 16/9.
• Round up to 85: the ratio becomes 150 / 85 ≈ 1.764706, off the other way by 0.7353 percent.

Neither is right, because the target does not exist at that size. Now do the same thing at a width of 1500: the exact height is 843.75, rounding to 844 gives 1500 / 844 ≈ 1.777251, and the error is 0.0296 percent. The absolute rounding is the same half-pixel either way; it is the size it is measured against that changed, so the relative distortion shrinks as the image grows.

The practical version: at thumbnail sizes, pick dimensions whose ratio is exactly achievable in whole numbers rather than rounding a proportion, and check the reduced ratio the tool prints rather than assuming the file kept the one you started with.