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What an angle is

Geometry · level 3, lesson 1 of 6

A question to start

Stand up and face the door. Now turn until you face the window. You did not walk anywhere, and yet something happened: you turned. How much did you turn? A little? A lot? Was it more or less than turning to face the wall behind you?

Here is a stranger question. Draw a small corner on a piece of paper, like the letter L. Now draw the same corner again, but make its two lines ten times longer. Which corner is bigger?

Most people say the long one. That is wrong, and by the end of this lesson you will know why.

The idea

An angle is an amount of turning

Put a pencil flat on the table. Hold one end still with your finger and swing the other end round. The pencil sweeps round like the hand of a clock. The amount it turns is called an angle.

So an angle is not a thing you can pick up. It is a measure of turning, the way a ruler measures length and a scale measures weight.

When we draw an angle, we draw where the turning started and where it stopped:

  • The two straight lines are called the arms of the angle.
  • The point where the arms meet is called the vertex (say "VER-tex"). It is the point that stays still, like your finger on the pencil.
  • The angle is the turn from one arm to the other. We often show it with a small curve or a coloured slice near the vertex.

The length of the arms does not matter. Only the turn matters.

short arms long arms: the same angle

Both coloured slices show the same turn. Making the arms longer is like making the clock hands longer: the hands of a tower clock are huge, but at 3 o'clock they make exactly the same angle as the hands on a wristwatch at 3 o'clock. That answers the question at the start: the two corners are the same size.

Turns you already know

You have known some angles for a long time without the name.

  • A full turn: you spin all the way round and face where you started.
  • A half turn: you turn round to face the other way, like "about turn!"
  • A quarter turn: half of a half turn, like turning from facing the front of the room to facing a side wall.

A clock shows them well. The minute hand makes a full turn every hour. From 12 to 3 is a quarter turn; from 12 to 6 is a half turn.

12 to 3: a quarter turn 12 to 6: a half turn

The five kinds of angle

Mathematicians sort angles into five families. They all compare the angle with two turns you already know: the quarter turn and the half turn.

acute right obtuse straight reflex

1. A right angle is exactly a quarter turn. It is the corner of a square, a sheet of paper, a book, a door frame. We draw a small square at the vertex to show it is a right angle, instead of a curve. The word "right" here does not mean "the opposite of left" or "correct". It comes from an old word meaning "upright", like a wall standing straight up from the floor.

2. An acute angle is smaller than a right angle. "Acute" means sharp, and acute angles look sharp: the tip of a slice of pizza, the point of a star, the beak of a bird.

3. An obtuse angle is bigger than a right angle but smaller than a half turn. "Obtuse" means blunt. A laptop screen tipped back, the arms of a reclining chair and the hands of a clock at 4 o'clock make obtuse angles.

4. A straight angle is exactly a half turn. The two arms point in opposite directions and make one straight line. It might not look like an angle at all, but it is: you turned, so there is an angle.

5. A reflex angle is bigger than a half turn but smaller than a full turn. "Reflex" means bent back. Reflex angles are harder to spot, because they go the long way round.

Every pair of arms makes two angles

Here is something many people miss. When two arms meet, there are two turns between them: the short way and the long way round. Picture the hands of a clock at 2 o'clock. The short way from the 12 to the 2 is a small acute angle. But you could also go from the 12 the other way round, past 11, 10, 9 and so on, all the way to 2. That long way is a reflex angle.

The two angles together make one full turn. So if the short way is small, the long way is big, and the other way round.

Usually when someone says "the angle", they mean the smaller one. If they mean the reflex one, they should say so, and the drawing should show the curve going the long way round.

A right-angle tester you can make

You do not need any tools to tell the families apart. Take any piece of paper and fold it in half, then fold it in half again the other way, so the two folded edges lie on top of each other. The corner where the folds meet is a perfect right angle. (Folding a straight edge onto itself makes a half turn into two equal quarter turns.)

Now hold your tester against any angle, with the vertex in the corner and one arm along an edge:

  • If the other arm is hidden under the paper, the angle is acute.
  • If it lies exactly along the other edge, it is a right angle.
  • If it sticks out past the edge, the angle is obtuse (or bigger).

Why turning matters

Angles are everywhere that things turn or meet. A steering wheel, a door, a skateboarder's spin ("a 180" is a half turn, "a 360" is a full turn, as you will see in the next lesson), the hands of a clock, the corners of a building. Builders need right angles so that walls stand up straight. Mars rovers are steered by telling them how much to turn, because nobody on Earth can grab the wheel.

Worked examples

1. Angles on a clock. What kind of angle do the hands make (the smaller one) at each time?

  • 3 o'clock: the hands point at 12 and 3. That is a quarter turn: a right angle.
  • 1 o'clock: the hands point at 12 and 1. That is much less than a quarter turn: acute.
  • 5 o'clock: from 12 to 5 is more than a quarter turn (12 to 3) but less than a half turn (12 to 6): obtuse.
  • 6 o'clock: the hands point opposite ways: a straight angle.

And the long way round at 1 o'clock, from 12 backwards past 11, 10, ..., to 1? That is more than a half turn but less than a full one: reflex.

2. Steering a robot. A toy robot faces north. It is told: "turn a quarter turn clockwise, then a half turn". Which way does it face now?

  1. North, then a quarter turn clockwise: north to east. It faces east.
  2. A half turn from east points the opposite way: west.

Check another way: a quarter turn plus a half turn is three quarters of a turn. Three quarter turns clockwise from north go north, east, south, west. The robot faces west. Both ways agree.

3. Angles in letters. Find the angles in the capital letters L, V and T.

  • L: the two strokes meet in a square corner. One right angle.
  • V: the strokes meet at the bottom in a sharp point. One acute angle. (The outside of the V, the long way round, is a reflex angle.)
  • T: the upright stroke meets the top bar in the middle. There are two right angles, one on each side, and together they make a straight angle along the top bar.

4. How many right angles fit? How many quarter turns make a full turn? Turn from north: east (1), south (2), west (3), north again (4). Four quarter turns make one full turn. And two quarter turns make a half turn. Keep these two facts: they are the key to the next lesson.

Try it

  1. Name the kind of angle: (a) the corner of a door; (b) the tip of a pencil; (c) the hands of a clock at 6 o'clock.
  2. What kind of angle (the smaller one) do the hands of a clock make at 4 o'clock? At 2 o'clock?
  3. You face south and make a half turn. Which way do you face?
  4. You face east and make three quarter turns clockwise. Which way do you face?
  5. Sam says: "My angle is bigger than yours because I drew longer lines." Is Sam right? Explain.
  6. The smaller angle between two arms is acute. What kind of angle is the bigger one, the long way round?
  7. How many half turns make a full turn? How many quarter turns make a half turn?
  8. A skateboarder spins two and a half full turns. How many quarter turns is that?
Answers
  1. (a) Right angle: a door corner is square. (b) Acute: the tip is sharp. (c) Straight angle: the hands point in opposite directions.
  2. 4 o'clock: from 12 to 4 is more than 12 to 3 (a quarter turn) but less than 12 to 6 (a half turn), so it is obtuse. 2 o'clock: less than a quarter turn, so acute.
  3. North. A half turn points you the opposite way, and the opposite of south is north.
  4. North. Clockwise from east: south (1), west (2), north (3).
  5. No. An angle measures the turn between the arms, not how long they are. Longer arms with the same turn make the same angle.
  6. Reflex. The two angles make a full turn together. An acute angle is less than a quarter turn, so the other one is more than three quarters of a turn, which is more than a half turn but less than a full turn.
  7. Two half turns make a full turn. Two quarter turns make a half turn.
  8. One full turn is 4 quarter turns, so two full turns are 8, and a half turn is 2 more: 10 quarter turns.

Common mistake

Judging an angle by the length of its arms. A big drawing does not mean a big angle. Look only at how far apart the arms open. If you are unsure, imagine shrinking the arms until they are tiny: the opening stays the same.

Forgetting the angle that goes the long way round. When two arms meet there are always two angles, and they add up to a full turn. Always check which one the question means: the curve in the drawing tells you.

Where it leads