How to calculate screen size
Every number on this site comes from one theorem — Pythagoras — used three ways. Here is each formula with a fully worked example you can check on paper.
1. Width and height from the diagonal
A screen is a rectangle, so width² + height² = diagonal². Knowing the aspect ratio w:h fixes the shape, and the diagonal then fixes the scale:
width = diagonal × w ÷ √(w² + h²)height = diagonal × h ÷ √(w² + h²)
Worked example — a 55″ 16:9 TV. √(16² + 9²) = √337 ≈ 18.358. Width = 55 × 16 ÷ 18.358 = 47.94 in. Height = 55 × 9 ÷ 18.358 = 26.96 in. Convert with the exact inch: × 2.54 gives 121.76 × 68.49 cm. Sanity check: 47.94² + 26.96² ≈ 55². ✓
2. PPI from resolution and diagonal
Apply the same theorem in pixel space. The diagonal pixel count is √(W² + H²); divide by the physical diagonal in inches:
PPI = √(W² + H²) ÷ diagonaldot pitch (mm) = 25.4 ÷ PPI
Worked example — a 27″ 4K monitor. √(3840² + 2160²) = √19,411,200 ≈ 4405.8 pixels. PPI = 4405.8 ÷ 27 = 163.18. Dot pitch = 25.4 ÷ 163.18 ≈ 0.156 mm — each pixel is about a sixth of a millimetre wide.
3. Diagonal from width and height
The direct direction: diagonal = √(width² + height²), in whatever unit you measured. A picture that is 47.94 in wide and 26.96 in high gives √(2298.2 + 726.8) = √3025 = 55 in — the 55″ TV from the first example, recovered exactly.
Common mistakes
- Using the resolution as the ratio without reducing: harmless in these formulas (the scale cancels), but 3440 × 1440 is not “21:9 exactly” — it simplifies to 43:18, which is why ultrawide dimensions computed from 21:9 are slightly off. Use the real resolution when you have it.
- Mixing units: keep the diagonal and the sides in the same unit before squaring. Convert first (1 in = 2.54 cm exactly), then calculate.
- Rounding mid-calculation: √337 rounded to 18 early costs you about a quarter inch at TV sizes. Carry full precision and round only the final answer — the calculators here do.
Frequently asked questions
Why does the formula need the aspect ratio at all?
The diagonal alone describes infinitely many rectangles — a long thin one and a near-square one can share it. The ratio picks the shape; the diagonal then sets the size of that shape.
Can I calculate size from width alone?
Yes, if you know the ratio: diagonal = width × √(w² + h²) ÷ w. With width and a 16:9 ratio, that is width × 1.1474. Without the ratio, no — width alone is not enough.
Do these formulas work for phones?
Yes — same geometry, taller ratios (about 19.5:9). The phone page applies them to real models; just use each model's actual resolution so the ratio is exact.