How accurate is this protractor?

Why there is no single accuracy figure

You will see browser protractors advertise a number like ±0.1°. Treat that with suspicion. An accuracy figure is only meaningful alongside the conditions it was measured under — how sharp the image was, how long the rays were, how carefully the points were placed. Quoted with none of that, it is decoration.

So this tool does not print one. Instead, every reading carries its own uncertainty, computed from the measurement you actually made.

How the angle is computed

Given a vertex v and two ray endpoints a andb, the tool forms the vectors u = a − v andw = b − v and evaluates:

θ = atan2(|u × w|, u · w)

That is a standard, well-conditioned formulation, accurate to roughly fifteen significant digits in double-precision floating point. It is not where your error comes from.

One implementation note, since it affects results people actually see: the more obvious formula acos(u · w / (|u| |w|)) isnot used, because it degrades badly for nearly-parallel rays and can return NaN once rounding pushes its argument past 1 — exactly when you are placing a second ray close to the first.

Where the error actually comes from

Point placement — usually the largest term

A deliberate click lands within roughly half a pixel of where you meant it. On a ray of length L pixels, half a pixel of sideways error at the endpoint tilts the ray by about 0.5/L radians. So short rays are imprecise rays, and that is the single biggest thing under your control.

Image resolution and blur

You can only place a point as precisely as the edge you are aiming at is defined. A soft, compressed, or low-resolution edge is several pixels wide, and your placement uncertainty grows to match.

Perspective — the one that catches people out

This is the error that does not shrink when you zoom in. A camera that is not square to the subject projects the scene at an angle, and projection does not preserve angles. Photograph a rectangular window frame from off to one side and its corners will measure something other than 90°, no matter how carefully you click. The tool cannot detect this and will not warn you about it.

The fix is at capture time: stand square to the plane you are measuring, centre the subject, and avoid wide-angle lenses, which curve straight lines near the frame edges.

Screen scaling

Browser zoom, device pixel ratio and retina scaling all change how many screen pixels an image pixel occupies. Angles are unaffected — an angle is scale-invariant. Lengths are not, which is why real-world lengths require the scale step.

What the ± figure means

Each reading shows an uncertainty derived from your two ray lengths and your current zoom:

δθ ≈ σ · √( 1/|u|² + 1/|w|² + 2/(|u|·|w|) )

where σ is the placement uncertainty of one point in document pixels — half a screen pixel divided by the zoom level, or considerably more for a fingertip on a touchscreen. The third term inside the root is the vertex: it is shared by both rays, so its error does not cancel between them. Leaving it out would understate the figure, and understating it is the one direction that would make this dishonest.

The displayed value is always rounded up, and never shown as zero.

How to get your best reading

1. Zoom in before placing points. The ± figure falls as you do, and you can watch it happen.
2. Drag the rays out long. Placing endpoints far from the vertex is the cheapest precision available.
3. Nudge with the arrow keys. Tab to a handle and use the arrows — hold Shift for single-pixel steps. No pointer gesture places a point as precisely, which is why this is an accuracy feature and not merely an accessibility one.
4. Use snapping only when the answer really is a round number.Snapping to 15° makes a reading tidy, not correct.
5. Start from a square-on, sharp image. No amount of care in the tool recovers a badly-taken photo.

When you should not rely on this

A measurement from a photograph is evidence, not proof. Confirm with a calibrated instrument when the result carries real consequences — structural or load-bearing work, anything safety-critical, legal or insurance documentation, or manufacturing tolerances.

This tool is also not a medical device. Do not use it for clinical measurement, diagnosis, or treatment decisions.

Checking the tool for yourself

A quick verification you can run in under a minute: draw a triangle with legs of 3 and 4 units along the horizontal and vertical. Its angles must read 90°, 36.9° and 53.1°. Or trace any closed shape with the chain tool — the results row shows the interior-angle sum, which must equal (n − 2) × 180°. If either check fails, something is wrong, and we would want to hear about it.

Open the tool · Read the FAQ