Math in Photography: Shutter Speed, Aperture, and Focus

I ruined the school play photos twice

My daughter was in the second row of a three-act play, and I brought the good camera. Halfway through the first song I looked at the screen and every moving child was a soft smear. I had shot the whole thing at 1/30 of a second in a dim hall, and 1/30 is the setting for a subject standing still.

So I fixed it the obvious way. I wound the aperture wide open to f/1.8, which let in far more light and let me speed the shutter up to 1/250. The blur went away. Something worse arrived instead: only the front row was sharp. I had bought shutter speed by selling focus.

That is the whole article. A camera has one real subject, light, and three dials that buy it. Every dial has a price, and the price is paid in something other than light. Once light has a price you can trade it, and the moment you can trade it you are doing arithmetic, not guessing. The interesting part is that the arithmetic is stranger than it looks: the numbers on the dial are not the amount of anything.

A stop is a doubling, and that is the whole trick

Photographers measure light in stops, and one stop is always a doubling or a halving. Not a little more. Not a percentage. Exactly twice, or exactly half. Halve the light and the picture gets darker but the shadows keep more detail. Double it and the shadows wash out. That trade is the reason the word exists.

Every setting on a modern camera is printed as a power of two, and the dials move in halvings. Shutter speeds run 1/60, 1/125, 1/250, 1/500. Sensitivity runs 100, 200, 400, 800, 1600. Aperture runs f/1, f/1.4, f/2, f/2.8, f/4. Each click halves the light or doubles it. Our powers and roots guide explains why a doubling is written 2 to the power n, and a camera is nothing but a row of those.

Here is the consequence, and it is the part that makes photography feel like arithmetic. If all three dials count in the same unit, then one stop on any dial trades exactly against one stop on any other dial. Widen the aperture one stop and you must close the shutter one stop to stay at the same brightness. Push sensitivity up one stop and you must give the light back somewhere. There is no default setting. There is only a budget you are balancing.

Aperture is an area, and the number is lying to you

Here is where it stops being obvious. The f-number is not the width of the hole. It is the width divided by the focal length, which is there so that two lenses with different sizes can produce the same picture. What actually gathers light is the area of the opening, and area goes as the square of the width.

So look at what happens between f/2 and f/2.8. The number went up by 0.8, which feels like nothing. The area went down by a factor of 0.51, which is almost exactly half. That is a full stop, and a full stop is a doubling or a halving. If you remember nothing else about apertures, remember that f/2.8 is not a slightly smaller version of f/2. It is half as much light, and it looks like nothing on the dial.

Go two stops and the square shows up cleanly: f/2 to f/4 is four times less light, exactly, because 2 times 2 is 4. The square root is why the sequence looks strange. Steps of about 1.4 rather than round numbers is what keeps every step an exact halving. If the area of a circle is the piece you are hazy on, our why pi times r squared page is worth five minutes first.

Shutter speed is secretly a frame rate

A shutter speed of 1/250 does not mean the shutter is open for 1/250 of the time you spend taking the picture. It means it is open for 1/250 of a second. Read it as a rate and the meaning flips: the fastest picture rate that setting supports is 250 frames per second, because a quarter of a second buys you 250 slots.

This is the ordinary habit of turning a unit into its reciprocal, which our rate and unit rate guide builds deliberately. It also explains why the two failure modes feel completely different. At 1/30 you are not taking an underexposed photo, you are taking a 30 fps video and hoping the motion stills itself. The arithmetic says the subject travels during the exposure. No amount of brightness fixes it, because blur and darkness are bought with different dials.

Focus is a tolerance, not a line

The third dial, focus, does not buy light at all, which is why it feels like a different subject from the other two. What it sets is how much blur counts as sharp. And that turns out to be a small piece of similar triangles: the size of the blur on the sensor is the focal length divided by the distance to your subject.

Push that one step and you get the only formula worth memorising in this article. Take the acceptable blur as about three hundredths of a millimetre, divide by the 43 mm diagonal of a full-frame sensor, and you get roughly 1/1400. Set focal length over that and the subject distance needed for sharpness comes out at about 1400 times the focal length. For a 50 mm lens that is 70 metres. Focus there and everything from 35 metres to infinity is acceptable, a single doubling of distance.

That is what widening the aperture really costs you, and it explains my ruined play photos exactly. At f/1.8 that band collapses to a few centimetres, and every row behind the first went soft. Our similar figures guide covers the triangle method if the ratio is new to you. Note what the tolerance changes: sharp is not a property of the subject, it is a property of the sensor and the screen you are judging it on.

What to try this week

Take one indoor photo of a moving subject and set the shutter to 1/15. Then take the same photo at 1/125 and at 1/1000, changing nothing else, and line the three up. Now do the same walk with the aperture fixed and the focus moved from your child to the far wall. Two rows of photographs, and every one of them is a fraction, an area, or a similar triangle. Nothing else is happening.

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