Angles are turns

Push open your classroom door and stop halfway. Now push it until it touches the wall. The door did not get longer, and it did not move to a different place: it turned around its hinges. How much it turned is what mathematicians call an angle.

Angles are hiding everywhere once you start looking. The hands of a clock make one at every moment. A pair of scissors makes one that grows as you open it. A cricketer's bat makes one with the ground, a slice of pizza makes one at its tip, and your elbow makes one every time you lift a glass of water.

In this layer you will learn to see angles, to compare them, to give them names and to measure turns in a special unit called the degree. No difficult symbols, just turning, folding and looking.

Look around you and find something that turns about a fixed point:

• A door turns about its hinges.
• The hands of a clock turn about the centre pin.
• Scissors open about the little screw in the middle.
• A ceiling fan blade turns about the motor at its centre.
• Your forearm turns about your elbow.

In every case there is a fixed point that does not move (the hinge, the pin, the screw, the elbow) and something that swings around it. The angle is the amount of swing.

A door that has opened a little has made a small angle. A door flung open against the wall has made a big angle. So even before we have numbers, we can already say which of two angles is bigger.

Here is a surprising idea. Two doors can be very different: one tall and heavy, one small like a cupboard door. If both are opened by the same amount of turn, they have made the same angle. The size of the door does not matter. Only the turn matters.

Hold that thought: it is the single most important thing to understand about angles, and it trips up many people much older than you.

Now let us draw an angle instead of turning one. Put your pencil on a dot. Draw a straight line out from the dot in one direction. Go back to the same dot and draw another straight line in a different direction.

You have drawn an angle. It has three parts:

• The vertex is the corner point where the two lines start (the dot). Say it VER-tex. The plural is vertices.
• The two straight lines are the arms of the angle.
• The opening between the arms is the angle itself.

Each arm starts at the vertex and goes off in one direction. In geometry, a straight path that starts at a point and goes on forever in one direction is called a ray (like a ray of sunlight from the Sun). So an angle is made of two rays that share the same starting point. You can learn more about rays in the lines topic.

An angle splits the flat page around it into two regions:

• The inside of the angle (its interior) is the space between the two arms, where the opening is.
• The outside (its exterior) is all the rest of the page.

Imagine a slice of pizza. The cheesy part between the two cut edges is the inside of the angle at the tip. The rest of the pizza, still on the plate, is outside that angle. The edges of the slice, the cut lines themselves, are the arms: they are neither inside nor outside.

Stand up, face the classroom door and turn all the way round until you face the door again. You have made a full turn (also called a complete turn, one revolution or one rotation).

Now try these:

• Turn until you face the wall behind you. That is a half turn. Two half turns make a full turn.
• From the start, turn until you face the wall on your right. That is a quarter turn. Four quarter turns make a full turn.
• A three-quarter turn takes you to the wall on your left, the long way round.

People have always needed to measure turns more exactly than "a bit" or "a lot". Thousands of years ago, astronomers in ancient Babylon (in today's Iraq) are thought to have started splitting a full turn into 360 equal small turns. We still do the same today. Each small turn is called one degree, written 1°. The little circle ° is the degree sign.

A quarter turn is so important that it has its own name: a right angle. It measures exactly 90°.

You see right angles all around you: the corners of your notebook, a window pane, a floor tile, a door frame, a cricket stump standing straight up from flat ground, the plus sign +. Builders care a lot about right angles, because a wall that is not at a right angle to the floor might lean and fall.

In drawings, a right angle is marked with a small square in its corner, instead of the curved arc we use for other angles. When you see that little square, you know the angle is exactly 90° without measuring.

Your right-angle checker can do more than find right angles. Hold it against any corner:

• If the corner fits inside your checker with space left over, the angle is smaller than a right angle.
• If it fits exactly, it is a right angle.
• If it is wider than your checker, the angle is bigger than a right angle.

This simple test sorts every angle into groups. The groups have names, which you will meet in the next chapter.

An angle smaller than a right angle (more than 0° but less than 90°) is called an acute angle. Acute comes from a Latin word meaning sharp, like the point of a needle. The tip of a pizza slice, the V of a bird's beak and the letter V are acute angles.

An angle bigger than a right angle but smaller than a straight line (more than 90° but less than 180°) is called an obtuse angle. Obtuse means blunt. A laptop screen tipped back to look at, a reclining chair, and the corner of a stop-sign shape are obtuse angles.

When the two arms point in exactly opposite directions and make a straight line, the angle is a straight angle: exactly 180°, a half turn. A book lying open flat on the table makes a straight angle along its spine.

What if you keep turning past a straight line? Open a door all the way round... well, doors cannot, but a clock hand can, and so can your angle-maker.

An angle bigger than a straight angle but smaller than a full turn (more than 180° but less than 360°) is called a reflex angle. Reflex angles are the "long way round".

Here is the trick. Whenever you draw two arms from a vertex, there are two openings: the small one on one side and the big one going round the other side. Together they make a full turn. If the small opening is 60°, the big opening is 360° − 60° = 300°, which is reflex. Usually when we say "the angle", we mean the smaller one, unless someone asks for the reflex angle.

• A complete angle (full angle) is exactly 360°: one full turn, ending where you started.
• A zero angle is exactly 0°: the two arms lie on top of each other and there is no turn at all.

Maps and compasses use four main directions: North (N), East (E), South (S) and West (W). If you stand facing north at sunrise, the Sun rises roughly on your right, in the east.

The four directions are spaced a quarter turn apart:

• From N to E is a quarter turn: 90°.
• From N to S is a half turn: 180°.
• From N all the way round to N again is a full turn: 360°.

The direction of turning matters too. Turning the same way as the hands of a clock is called clockwise. Turning the other way is anticlockwise (in some countries, counterclockwise). Facing north, a quarter turn clockwise brings you to east; a quarter turn anticlockwise brings you to west.

A clock is a wonderful angle machine. The 12 numbers are spread evenly all the way round, so the gap from one number to the next is:

360° ÷ 12 = 30°.

This gives you an easy way to find the angle between the hands on the hour: count the numbers between the hands and multiply by 30.

• At 1:00 the hands are 1 number apart: 1 × 30° = 30° (acute).
• At 2:00: 2 × 30° = 60° (acute).
• At 3:00: 3 × 30° = 90° (right angle).
• At 4:00: 4 × 30° = 120° (obtuse).
• At 6:00: 6 × 30° = 180° (straight angle).
• At 12:00: the hands overlap: 0° (zero angle).

The minute hand is quicker. It goes all the way round (360°) in one hour, so in 15 minutes it turns a quarter turn, 90°, and in 30 minutes a half turn, 180°.

Sometimes two angles sit side by side and together make something special.

• Two angles that together make a right angle (90°), like 30° and 60°, are called complementary angles. Think of them as two pieces that complete a square corner.
• Two angles that together make a straight angle (180°), like 120° and 60°, are called supplementary angles.

Look at a road junction where a straight road meets a side road. The side road splits the straight line into two angles. However the side road is tilted, the two angles always add up to 180°, because together they make the straight line. That means if you know one of them, you can work out the other without measuring!