Angles

An angle tells you how much something turns. A door that opens a little makes a small angle with the wall. A door that opens wide makes a big one. We measure angles in degrees, written with a small circle: 90° means "90 degrees". The idea of cutting one full turn into 360 parts is very old. It probably comes from the astronomers of ancient Babylon, more than 2,000 years ago. The number 360 is a good choice, because it can be divided exactly by 2, 3, 4, 5, 6, 8, 9, 10, 12 and many more numbers. With just three numbers to remember, 90, 180 and 360, you can find almost any missing angle.

What an angle is

An angle is made when two straight lines meet at a point. The point where they meet is called the vertex. The size of the angle is the amount of turn from one line to the other. It does not depend on how long the lines are drawn. If you make the two lines longer, the angle stays the same.

Angles also have names that tell you their size:

• An acute angle is smaller than 90°. The tip of a slice of pizza is an acute angle.
• A right angle is exactly 90°. It is the corner of a square, a book or a sheet of paper.
• An obtuse angle is between 90° and 180°. A door that is opened wide makes one.
• A straight angle is exactly 180°. Its two lines make one straight line.

The three numbers to know by heart

Each number is built from the one before. Two right angles make a straight line: 90° + 90° = 180°. Four right angles make a full turn: 4 × 90° = 360°. That is also why the four corners of a rectangle add up to 360°: each corner is a right angle, and there are four of them.

Two more facts are worth knowing here. The three angles of any triangle add up to 180°. So if all three angles are equal, each one is 180° ÷ 3 = 60°. There is a whole lesson on angles in a triangle.

Angles on a straight line

Draw a straight line. Now draw a ray that starts at a point on the line and goes up. (A ray is a line that starts at a point and goes on in one direction.) The ray cuts the straight angle of 180° into two parts. The two parts always add up to 180°.

1. Find the angle you know.
2. Subtract it from 180°.
3. Check: the two angles together must make 180°.

Two angles that make a right angle

Sometimes two angles fit together to make a right angle, like two pieces of a square corner. Then they add up to 90°. Two angles like this are called complementary angles.

Angles around a point

When several angles sit around one point and fill the whole space around it, they make one full turn. Together they add up to 360°. Think of the slices of a round cake: all the slices meet in the middle, and together they go all the way round.

Which total do I use?

Most angle questions are the same job: find the right total, then take away the angles you know. This table shows which total goes with which picture.