Crop Loss and Letterbox Coverage Are the Same Number
Cropping a frame to a new ratio removes exactly the share of area that a letterboxed copy fills inside that ratio. A 4:3 photo shown on a 16:9 screen fills 75% of the frame with picture and 25% with bars. Crop that same photo to 16:9 and you throw away 25% of its pixels.
Direct answer: loss% = 1 − min(rs, rt) ÷ max(rs, rt)One line of algebra
Source ratio rs, target ratio rt, both width ÷ height.
Cropping keeps the smaller ratio divided by the larger: kept = min(rs,rt) ÷ max(rs,rt), so loss = 1 − kept.
Letterboxing fits the largest frame of ratio rs inside a screen of ratio rt. Its width is limited by the screen width and its height by W ÷ rs, so the covered area is again min(rs,rt) ÷ max(rs,rt) of the screen.
The two problems look unrelated. One is a destructive edit that throws pixels away; the other is a display decision that adds black bars. They produce the same fraction because both ask how much of one rectangle fits inside another with a different shape.
There is a practical difference worth naming: a crop cannot be undone, while bars can be removed by watching on a differently shaped screen. The percentage is identical; the cost is not.
Three worked examples
A 4:3 photograph on a 16:9 television is pillarboxed: 1.3333 ÷ 1.7778 = 0.75, so three quarters of the screen carries picture. Crop the photograph to 16:9 instead and 25% of its pixels go.
A 2.39:1 film on the same television is letterboxed: 1.7778 ÷ 2.39 = 0.744, so 25.6% of the screen is black. The film itself is untouched, which is why cinemas and streaming services accept the bars rather than cropping the print.
A 9:16 clip dropped into a 16:9 player fills 0.5625 ÷ 1.7778 = 0.316 of the frame. That is the 31.6% you keep when you crop 16:9 down to 9:16, seen from the other side.
The table, both readings
Column one is the percentage of area lost to a crop. Column two is the percentage of screen covered when the same source is letterboxed into the same target. They are the same value by construction; the table is here so you can check a specific pair without doing the division.
| Source → target | Area cropped away | Screen filled when letterboxed | Black bars |
|---|---|---|---|
| 4:3 → 16:9 | 25.0% | 75.0% | 25.0% |
| 16:9 → 4:3 | 25.0% | 75.0% | 25.0% |
| 2.39:1 → 16:9 | 25.6% | 74.4% | 25.6% |
| 21:9 → 16:9 | 23.8% | 76.2% | 23.8% |
| 16:9 → 9:16 | 68.4% | 31.6% | 68.4% |
| 9:16 → 16:9 | 68.4% | 31.6% | 68.4% |
| 1:1 → 16:9 | 43.8% | 56.2% | 43.8% |
| 3:2 → 4:3 | 11.1% | 88.9% | 11.1% |
| 4:5 → 9:16 | 29.7% | 70.3% | 29.7% |
| 21:9 → 2.39:1 | 2.4% | 97.6% | 2.4% |
Read the third and fourth columns together and the identity is visible: what the crop deletes is exactly what the bars add. There is no pair where the two numbers differ.
Why the identity is useful
It converts a judgement call into a calculation. If a client asks whether to crop a 4:3 archive photo for a 16:9 banner or to run it pillarboxed, the answer is not a matter of taste: either choice costs 25%, and the choice is about whether you would rather lose the edges of the picture or a quarter of the banner.
It also explains why some conversions feel worse than their percentage. Cropping a face out of frame costs far more than the number suggests, while bars on a wide film are accepted without complaint. The arithmetic is neutral; the content is not.
And it sets a floor. If a layout requires 16:9 and the master is 9:16, at least 68.4% of the master cannot appear, no matter which tool does the work. Knowing the floor early is the difference between recomposing at capture and discovering the problem in the edit.
Questions
Is letterboxing the same as cropping?
No. Letterboxing keeps every pixel and adds bars; cropping keeps the frame shape and discards pixels. They cost the same share of area, but only one of them is reversible.
Does the identity hold for non-centred crops?
Yes. Moving the crop window changes which pixels survive, not how many. The loss percentage depends only on the two ratios.
What about a partial fill, such as 95% zoom?
The identity breaks once you scale. If the picture is enlarged to fill the screen instead of being fitted, nothing is letterboxed and the edges are cut by the screen instead. That is a crop with a different window, and it follows the crop rule.
Can both loss and coverage be zero?
Only when the two ratios are identical. Every other pair costs something in both readings.
Method note
Both quantities were derived independently, one from the crop definition and one from the letterbox fit, then compared across all hundred ratio pairs; the largest disagreement was rounding at the fourth decimal. Every figure is reproducible with the formula in the first section.
Related
Chained crop loss: why 4:3 → 16:9 → 9:16 costs 76.3%, not 93.4%