Gear teeth are that shape for a reason

The involute curve, and the property that makes it work.

A gear tooth carries that specific curved profile, an involute, rather than a simple triangle or a plain curve someone found convenient, because that one particular shape is what lets two meshing gears turn at a perfectly steady speed ratio relative to each other throughout the entire time each pair of teeth is in contact, and almost no other shape manages that.

Taking apart a dead cordless drill for the first time and finding three separate sets of gears stacked inside, where a single set looked, at a glance, like it should have done the whole job alone, was the moment gear teeth stopped looking like an arbitrary shape and started looking like something worth actually understanding properly.

What is really going on

Two gears meshing together only transmit motion smoothly if the speed ratio between them stays exactly constant while each pair of teeth passes through contact, since any wobble in that ratio, even a slight one repeated every time a new tooth engages, shows up as vibration, noise and uneven wear rather than a clean, steady transmission of motion. An involute curve has the specific mathematical property that the line connecting the point of contact between two meshing teeth always stays tangent to two fixed circles, one on each gear, no matter which exact point along the curve happens to be in contact at that instant, and it is precisely this property that keeps the speed ratio locked constant throughout the whole engagement rather than only at one favoured moment.

The unwound-string comparison

Tying a piece of string to a spool, keeping it taut, and slowly unwinding it while tracking the path traced by the string's free end draws exactly the curve a gear tooth's profile is built from, an involute, the shape a taut line traces as it peels away from a circle at a constant rate. Every point along that traced path sits at a slightly different distance from the spool, yet the string itself, at every single instant, remains tangent to the spool's edge, which is the same geometric fact that keeps two meshing involute teeth's line of contact tangent to their own underlying circles throughout meshing, holding the speed ratio steady even as the actual contact point slides along each tooth's face during engagement.

Why a simpler-looking tooth shape does not work

A tooth cut to a plain curve chosen for looking reasonable, rather than derived from this specific geometric construction, meshes correctly at exactly one instant, wherever it happened to be designed to fit perfectly, and drifts away from a constant speed ratio everywhere else in its engagement, producing a small but real speeding-up-and-slowing-down effect repeated on every single tooth. That repeated tiny fluctuation is exactly what shows up as noise and vibration in a poorly profiled gear, since a driven shaft being sped up and slowed down many times per revolution, however slightly each time, has to be felt and heard somewhere in the mechanism carrying it. The involute curve is not an aesthetic choice or a manufacturing convenience, it is specifically the one family of curves that avoids this fluctuation entirely across the whole range of contact, which is why it became the standard tooth profile almost universally rather than one option among several equally good ones.

The number that matters here

An involute gear's tooth profile forgives small errors in how far apart the two gears' centres are actually mounted, keeping the constant-speed-ratio property intact even if the real centre distance differs from the design value by a modest amount, a tolerance that a tooth shape without this property could not offer at all, since almost any other curve's correct behaviour depends on the centre distance being held to the exact value it was designed around. This forgiveness is a large part of why involute gears are practical to manufacture and assemble at ordinary tolerances, rather than needing the near-perfect precision a less forgiving tooth profile would otherwise demand, a genuinely practical concern in any gearbox where shafts, bearings and housings each contribute their own small share of positioning error before the two gears ever actually meet.

What this changes in practice

Recognising why the involute shape exists changes how a damaged or worn gear tooth gets judged, since even a small amount of wear or chipping distorts the exact curve the tooth needs to maintain that constant contact property, and a gear that still looks broadly tooth-shaped to the eye can already be transmitting motion unevenly well before it looks obviously damaged. It also explains why replacement gears have to match a specific, standardised tooth profile rather than merely a similar-looking outline, since two gears built to visibly similar but mathematically different curves will mesh badly together even if each one, inspected alone, looks entirely reasonable. This is also why cutting a one-off replacement gear by eye, matching only the tooth count and the rough outline of the original, so often produces a part that turns yet runs noticeably rougher than the gear it replaced, the missing ingredient being the exact curve rather than anything visible in a casual comparison of the two.

Where this stops being true

The involute profile solves the problem of a constant speed ratio during meshing, it says nothing on its own about noise from other sources, backlash between teeth, imbalance in the gear body, or resonance in the shaft carrying it, all of which can still make a perfectly cut involute gear pair sound rough despite the tooth profile itself doing exactly what it was designed to do. A quiet, smooth-running gear pair needs the involute geometry to be correct and several other things besides it, and the tooth shape alone is only ever one part of that larger picture, a distinction this set returns to directly once it reaches the specific question of where gear noise actually comes from.