Online Protractor

Measuring an angle on your screen

An online protractor behaves much like the plastic one from school, with one advantage: nothing slips. Drag either arm and it follows your pointer, snapping to the nearest half degree while the reading updates. To measure something real, hold it flat against the display with its corner on the vertex mark at the centre of the baseline, then bring each arm onto one of the two edges.

There is a second mode people often actually want, which is drawing a specific angle rather than measuring an unknown one. Type the value into the exact-angle box and the arms move to it, so you can hold paper against the screen and trace a true 45° or 30°. Switch on 15° snapping and dragging lands cleanly on the angles that come up most: 15, 30, 45, 60, 90 and so on.

Why no calibration is needed here

The ruler and the actual size viewer both need to know your display's pixel density before you can trust them, because both report physical lengths. A protractor does not, and the reason is worth stating: an angle is a ratio rather than a length. Scale a figure up or down and every angle inside it stays exactly the same. So this tool is right on any screen, at any resolution, with no setup.

What does limit precision is your eye. The readout resolves to a tenth of a degree, finer than you can read off a moulded plastic scale, so in practice the accuracy you get depends on how carefully you align the arms to the edges in front of you. If you want more room to do that, the fullscreen button roughly doubles the radius, which means a degree spans twice the arc length.

Degrees, radians and naming angles

Degrees divide a full turn into 360 parts. Radians measure the same turn as 2π. Radians are the natural unit in trigonometry and in most programming languages' maths libraries, which is why both appear. The conversion is degrees × π/180, so a right angle is π/2, or roughly 1.571 rad.

The vocabulary is worth having straight, since it is usually what a homework question is really testing:

• Below 90° an angle is acute.
• At exactly 90° it is a right angle.
• Between 90° and 180° it is obtuse.
• At 180° it is a straight angle.
• Beyond 180° it is reflex.

Because a semicircular protractor only reads to 180°, whether plastic or on screen, a reflex angle is found by measuring its smaller partner and subtracting from 360. That figure sits beside the main reading, so the arithmetic is done for you.

Reading the right scale

A moulded protractor carries two rings of numbers running in opposite directions, and picking the wrong one is the single most common mistake made with it. An angle read as 130° on the outer ring is 50° on the inner. The rule that settles it never changes: decide first whether the angle is smaller or larger than a right angle, then choose the ring whose number agrees with you. An angle that plainly looks narrow cannot be 130°, whatever the ring under your pencil says.

The version on this page has one scale, so the question does not arise. It also reads to a tenth of a degree, finer than the half-degree divisions moulded into plastic, and the arms stay where you put them instead of drifting while you reach for a pencil. For the full reasoning behind the two scales, vertex alignment and reflex angles, see how to read a protractor.

Finding an angle without a protractor

Sometimes the shape will not come to the screen. A few methods need nothing but a ruler:

• From two side lengths. In a right-angled triangle, measure the side opposite your angle and the side beside it, divide the first by the second, and take the inverse tangent. A rise of 30 mm over a run of 52 mm gives 0.577, which is 30°.
• By folding. Fold a square corner of paper in half and you have 45°; fold that again for 22.5°. Paper is cheap and the fold is exact.
• The 3-4-5 triangle. Mark 300 mm along one edge and 400 mm along the other; when the diagonal between the marks measures exactly 500 mm, the corner is square. Builders have squared walls this way for a very long time.

The angles that come up most

Homework is the obvious one, and the vocabulary above is usually what a question is really testing. Beyond that the requests repeat. A mitre joint needs each piece cut at half the corner angle, so a 90° corner takes two 45° cuts, and a corner that is not quite square takes two cuts that are not quite 45°, which is why measuring it first matters. Roof pitch is quoted as a rise over a run rather than in degrees, and the inverse tangent above converts one to the other. Sewing patterns give darts in degrees. A tripod head, a drafting board, a satellite dish and a wheelchair ramp all specify angles, and a ramp in particular has a legal maximum in most building codes.

For anything you can hold flat against the display, the method is the same each time. Put the corner of the object on the vertex mark at the middle of the baseline, line one edge up with the baseline itself, and swing the other arm onto the second edge. Full screen roughly doubles the radius, and a longer arm is a more forgiving one: the same misjudgement of a millimetre at the tip costs you half as much of a degree.