Online protractor

Upload a photo or measure straight on the screen: drag the arms and read the angle in degrees and radians.

Upload a photo or a screenshot and measure the angle on top of it. The image never leaves your browser.

Choose image or drop one here

Snap to

Reflex angle Long arms

Degrees: 59.0°
Radians: 1.030 rad
Kind: Acute
Complement: 31.0°
Supplement: 121.0°

The angle is read off the image as it sits on screen. If the photo was taken from an angle, the angle in the photo is not the angle of the real object.

Generated right here, in your browser. What you type never leaves your device.

Keep going with Online ruler Printable paper Screen test Monitor calibration

A protractor is the one drawing instrument nobody keeps to hand. Rulers turn up in drawers, tape measures live in toolboxes, but the half-disc of clear plastic went back to school with somebody in about 2006 and has not been seen since. Which is a nuisance, because the moment you need to know whether a corner is square or a roof pitches at 30 degrees, you need it precisely then.

This one works two ways. Drag the arms around the dial to read an angle directly, the way the plastic one works. Or upload a photo — a corner of a room, a rafter, a diagram, a screenshot of a CAD drawing — and lay the protractor over it to measure an angle that already exists in the picture.

The photo never leaves your device. It is read straight into the page from your own disc, drawn to the screen, and forgotten when you close the tab. There is no upload, no account, and nothing to delete afterwards.

Manual

How to use it

1
Add a picture, or do not

Choose an image or drag one onto the box if you want to measure something in a photo. Skip it entirely if you just need a protractor on a blank field, which is what most people are after.
2
Put the vertex on the corner

Drag the middle handle to the point where the two lines meet. On a photo, this is the corner you are measuring; the whole reading depends on getting it in the right place.
3
Lay each arm along a line

Drag the two outer handles until each arm runs along one of the lines forming the angle. The arms keep their length as you swing them, so they stay easy to grab.
4
Extend the arms when a line runs long

Long arms continues each arm past its handle to the edge of the surface. On a photograph it lets you line an arm up with a wall or a rafter that carries on well beyond where the handle can sit, which is where most of the placement error otherwise comes from.
5
Centre it, or take the whole screen

The toolbar has a centre button that moves the whole figure back to the middle without changing the angle, for when a drag has walked it into a corner. Beside it, fullscreen gives the protractor the entire display — which is what you want with a photograph under it.
6
Read the angle

The figure carries its own reading, written on the bisector between the arms, and the panel underneath repeats it with the same value in radians and the name of the kind of angle it is. Complement and supplement appear where they exist, which saves the mental arithmetic when you are checking a joint.
7
Snap if you want round numbers

Set the snap to 1, 5, 15 or 45 degrees when you are constructing an angle rather than measuring one. Leave it off when measuring, because a reading that quietly rounds itself is worse than one that says 44.6.
8
Switch to the reflex reading

Tick the reflex box to read the angle the long way round, past 180 degrees. Useful for the outside of a corner, or for anything where the interesting angle is the one you are not standing in.

In depth

Things worth keeping in mind

Measuring angles in photographs

A camera flattens three dimensions into two, and that flattening changes angles unless the plane you are measuring is square to the lens. It is worth knowing when the number can be trusted.

Trustworthy: a photograph taken straight on, with the surface parallel to the camera back. A scanned drawing, a screenshot, a diagram.
Trustworthy: anything where you only need to compare two angles in the same photo, since both are distorted the same way.
Suspect: a corner of a room shot from across the room. Perspective will make a 90-degree corner read as anything from 60 to 120 depending on where you stood.
Suspect: wide-angle and phone ultra-wide lenses, which bend straight lines near the edges of the frame before any perspective is considered.
Fix it by moving: stand square to the surface and centre it in the frame. The further from the middle of the picture, the more the lens has moved things.

Angle names worth knowing

The tool names the angle for you, but the vocabulary comes up constantly in instructions, and knowing it makes the rest of a manual readable.

Acute: under 90 degrees. Right: exactly 90. Obtuse: between 90 and 180.
Straight: exactly 180, which is to say a line. Reflex: more than 180 and less than 360.
Complementary: two angles adding to 90. Cutting a corner joint, the two mitres are complementary.
Supplementary: two angles adding to 180. Where a line crosses another, the angles on one side are supplementary.
Bisect: to halve an angle. A mitre is the bisector of the corner two pieces meet at, which is why a 90-degree corner takes two 45-degree cuts.

History

How the circle got 360 pieces

The 360-degree circle comes from Babylon, and it survives because of two accidents that reinforced each other. The Babylonians counted in base 60, probably by using the thumb to tick off the three joints on each of the four fingers, and 360 is six sixties. It is also close to the number of days in a year, so a sky that turns roughly one degree a night was a convenient way to keep a calendar.

What made it stick was divisibility. 360 divides cleanly by 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120 and 180. No number near it comes close to that, and it is why a third of a circle is a whole 120 degrees while a third of a 100-unit circle would be 33.33 recurring.

That last point was tested. Revolutionary France, having decimalised weight, length and the calendar, decimalised the circle too: 400 gradians, a right angle of exactly 100. It was mathematically tidier and it failed anyway, because every existing table, instrument and trained surveyor spoke degrees. Gradians survive in some European surveying and on the "GRAD" button that nobody presses on a scientific calculator.

The word protractor is younger than the instrument. Semicircular scales for setting out angles were in use by mediaeval surveyors and navigators, but the name — from the Latin for "to draw out" — appears in English around 1610, in the context of mapmaking. The plastic school version is a twentieth-century invention, and it is the reason most people can picture one and almost nobody owns one.

360: degrees, inherited from base-60 Babylon
24: whole divisors of 360, which is why it survived
400: gradians in the circle France tried to standardise
1610: the word protractor first written in English

How it works

Two rays, one arctangent

Everything on the dial comes from one function. For each arm, the tool takes the difference between the arm handle and the vertex — a horizontal distance and a vertical one — and hands the pair to the two-argument arctangent. That returns the direction of the arm as an angle from due east. Subtract one arm’s direction from the other and you have the angle between them.

The two-argument form matters. Plain arctangent takes a slope and cannot tell the difference between a line going up-right and one going down-left, because both have the same ratio. Giving it the two distances separately preserves which quadrant the arm is in, so the answer is right across a whole turn rather than only across a half.

One sign has to be flipped along the way. On a screen the y coordinate grows downwards, which is inherited from how cathode-ray tubes painted a picture: top-left to bottom-right, a line at a time. Mathematics has y growing upwards. So the vertical distance is negated before the arctangent sees it, and the dial then turns anticlockwise from the right, which is what a protractor printed on paper does.

The last decision is which of the two angles to report. Two rays from a point always make a pair that adds to a full turn — a 60-degree corner is also a 300-degree corner from the other side. The tool shows the smaller by default, because that is what somebody measuring a mitre wants, and offers the reflex reading as a switch rather than guessing.

By hand

Measuring an angle with no protractor at all

This is worth knowing because it works on paper, on a wall, and on a photo — and because it is how angles were measured for the two thousand years before anyone thought to print a scale on a half-disc.

1

From the corner, measure the same distance out along each of the two lines and mark the spots. Ten centimetres is convenient. The distance does not matter as long as it is the same on both.
2

The straight distance from one mark to the other. Call it the chord, and the distance you used along the lines the radius.
3

That gives the sine of half your angle. If the chord is 7 cm and the radius 10 cm, that is 3.5 ÷ 10 = 0.35.
4

The inverse sine of 0.35 is about 20.5 degrees, so the angle is roughly 41 degrees. Any calculator does the inverse sine; a table of sines does it just as well and is how it was done for centuries.
5

A chord longer than the radius means the angle is over 60 degrees; a chord equal to the radius is exactly 60, because the triangle is equilateral. That single fact catches most arithmetic slips.

Diagnosis

If something is not working

Symptom:The angle jumps as I drag

Likely cause:: Snapping is on, so the arm is landing on whole multiples of the step rather than following the pointer.
What to do:: Set the snap to "No snap" while you are measuring. Snapping is for constructing an angle you already know, not for reading one you do not.

Symptom:It reads 300 when I expected 60

Likely cause:: The reflex switch is on, so you are reading the angle the long way round.
What to do:: Untick the reflex box. The two readings always add to 360, so if one is 300 the other is 60 and the tool is not wrong, just answering the other question.

Symptom:A square corner in my photo does not read 90

Likely cause:: Perspective. The camera was not square to the surface, so the corner is foreshortened in the picture.
What to do:: Retake the photo standing directly in front of the surface, with the corner near the middle of the frame. Nothing done afterwards can recover the true angle from a skewed shot.

Symptom:The handles are hard to grab on a phone

Likely cause:: A fingertip covers a bigger area than a cursor, and the three handles sit close together when the angle is small.
What to do:: Drag the vertex away from the arms first to spread them out, or open the angle wider, set the arms, then move the vertex back into place.

Symptom:My image looks squashed

Likely cause:: The picture is fitted inside the box without cropping, so a very wide or very tall image leaves empty space rather than being stretched.
What to do:: Nothing is distorted — the aspect ratio is preserved deliberately, because stretching it would change every angle in the picture, which is exactly what a protractor must not do.

Trivia

Things worth knowing

A right angle is only a quarter turn by convention

Ancient Egyptian rope stretchers laid out square corners with a loop knotted into twelve equal parts, forming a 3–4–5 triangle. They needed no unit of angle at all — the right angle came out of the lengths — which is why the technique works identically whether you call the corner 90 degrees, 100 gradians or a quarter turn.

The degree symbol was once a zero

The little raised circle comes from the abbreviation for "gradus", written in mediaeval manuscripts as a superscript o. The same lineage gives the minute and second marks used for fractions of a degree, still standard in navigation and astronomy: 60 minutes to a degree, 60 seconds to a minute, straight out of Babylon.

Roof pitch is usually not stated in degrees

Builders in Britain and the United States quote a rise over a run — "4 in 12" — rather than an angle, because it is what a framing square reads directly. A 4-in-12 pitch is 18.43 degrees. Measure a rafter with this and you will get the degrees; divide the rise by the run to speak to a roofer.

Gradians still hide in your calculator

The DEG / RAD / GRAD selector on a scientific calculator offers all three because gradians never entirely died. Surveyors in parts of Europe still use them, and 100 gradians to a right angle makes slope percentages fall out easily — a 100% slope being exactly 50 gradians, or 45 degrees.

Radians are the unit with no unit

A radian is the angle subtending an arc equal in length to the radius, so it is a ratio of two lengths and is technically dimensionless. That is why calculus is done in radians: the derivative of sine is cosine only if the angle is measured that way. In degrees a stray factor of π/180 turns up in every result.

Questions

Frequently asked questions

Does my photo get uploaded anywhere?

No. The file is read directly by the page from your own device and drawn to the screen. There is no server to send it to, no account, and nothing stored: closing the tab is all the deleting there is to do.

How accurate is it?

The arithmetic is exact. The accuracy is entirely in how precisely you place the three handles, which on a desktop is about a fifth of a degree with care. On a photo, perspective error is usually far larger than placement error.

What is the reflex angle for?

Reading a corner from the outside. If you are measuring the inside of a room corner you want the smaller angle; if you are cutting trim that wraps around the outside of the same corner, the reflex reading is the one you need.

Why does it show radians as fractions of π?

Because π/4 is the answer somebody working in radians actually wants, and 0.785 rad is the same number in a form they would then have to convert back. Where the angle is not a tidy fraction of π, the decimal is shown instead.

Can I measure an angle bigger than 180 degrees?

Yes, with the reflex switch. The arms themselves only ever make an angle up to 180 between them, so anything beyond that is the same corner read the other way round.

Does it need calibrating like the ruler does?

No, and this is the useful difference between the two. An angle is a ratio of lengths, so it survives any scaling: a corner is 30 degrees whether the screen renders it large or small. Only lengths need a physical reference.

Does it work on a phone or tablet?

Yes. The handles respond to touch exactly as they do to a mouse. Photos taken on the same phone can be loaded straight from the camera roll.

Can I use it for schoolwork?

It reports the same thing a plastic protractor does, and the snap setting makes constructing a given angle straightforward. Do check whether the exercise expects you to demonstrate using a physical instrument, because that is often the point of the task.

How to use it

Add a picture, or do not

Put the vertex on the corner

Lay each arm along a line

Extend the arms when a line runs long

Centre it, or take the whole screen

Read the angle

Snap if you want round numbers

Switch to the reflex reading