Every time you open a door, tip a seesaw, or read the hands of a clock, you are looking at an angle — a measure of how far something has turned. Learning to measure angles lets you say exactly how wide a corner or a turn is, instead of just “a little” or “a lot”.

An angle is the amount of turn between two rays that start from the same point. That shared point is the vertex, and the two rays are the arms of the angle. Keep one arm still and swing the other: the further you swing it, the wider the opening — and the bigger the angle.

An angle is about turning, not distance. It does not matter how long you draw the arms — a short pair of arms and a long pair of arms can hold exactly the same angle. What we measure is how much one arm has spun away from the other.

We measure that turn in degrees, written with a little circle: ^\circ. One full spin all the way around — back to where you started — is 360°. From there, every other angle is a fraction of a full turn:

• a full turn (once around) is 360^\circ;
• a half turn (a straight angle) is 180^\circ;
• a quarter turn (a right angle) is 90^\circ — the square corner, marked with a small square instead of an arc;
• a three-quarter turn is 270^\circ.

The two hands of a clock meet at the centre — that is the vertex — and the hands are the arms. At 3 o'clock the hands make a neat square corner: a 90^\circ right angle. At 6 o'clock they point opposite ways and lie flat — a 180^\circ straight angle. At 12 o'clock the hands sit on top of each other, so the angle is 0^\circ: no turn at all. Watch the hands all day and you will see every angle there is.

Drag the slider to swing the second arm around the vertex. Watch the reading climb from a thin sliver, through the right angle and the straight angle, all the way to a full turn.

Naming angles by size

Once you can measure an angle, you can sort it into a family by how big it is:

• Acute — smaller than a right angle: less than 90^\circ.
• Right — exactly 90^\circ (the square corner).
• Obtuse — between 90^\circ and 180^\circ.
• Straight — exactly 180^\circ, a flat half turn.
• Reflex — bigger than a straight angle: more than 180^\circ (but less than a full 360^\circ).

A quick way to remember the first two: an a-cute little angle is small and sharp, while an obtuse angle is big and blunt. Step through one of each below.

The pointy tip of a pizza slice is a vertex, and the two straight cuts are the arms. A thin, mean little slice has a small, acute angle. If you are hungrier and cut a fat slice — more than a square corner's worth — the tip opens up into an obtuse angle. Cut the whole pizza into eight equal slices and each tip is a 45^\circ angle, because 360^\circ \div 8 = 45^\circ.

Have a go: name the angle

Here are two arms making a fresh angle each time. Read its size in degrees, then decide which family it belongs to — acute if it is under 90^\circ, a right angle at exactly 90^\circ, or obtuse once it grows past a square corner. Press Refresh for a new one.

Reading a protractor

A protractor is a half-circle of plastic with the numbers 0 to 180 printed around its edge. To measure an angle with it:

1. Put the protractor's little centre cross right on the vertex.
2. Line one arm up along the baseline, pointing at the 0.
3. Follow the other arm out to the edge and read the number it crosses.

Three quick worked examples (the swing-arm above is a protractor laid out flat):

• One arm sits on the 0 and the other reaches the 50 mark, so the angle is 50^\circ — under 90^\circ, so it is acute.
• The other arm reaches the 130 mark, so the angle is 130^\circ — between 90^\circ and 180^\circ, so it is obtuse.
• The arm lands exactly on 90 and the corner looks like a perfect square: that is a 90^\circ right angle.

See it explained