Angles

At 3 o'clock the hands of a clock open into a perfect right angle; at 6 o'clock they lie flat into a straight line. How far the hands open is exactly what an angle measures.

An angle is made of a vertex and two rays starting from it. What it measures is not the length of the sides and not the space between them — it is how wide the two rays are spread apart.

A full turn is 360°, half a turn is 180°, a quarter turn is 90°. Keep those three benchmarks in your head and most everyday angles can be estimated at a glance.

Degrees and radians: two scales

The everyday unit for angles is the degree (written °): split a full circle into 360 equal parts and each part is 1°. Why 360? Blame the Babylonians — 360 divides evenly by 2, 3, 4, 5, 6 and many other numbers, which makes cutting it up convenient.

There is a second unit called the radian: half a turn (180°) corresponds to $\pi$ radians, so $90° = \frac{\pi}{2}$ . For now, remembering $180° = \pi$ is plenty — it will shine in high school.

Never judge by looks

An angle's size depends only on how wide it opens, not on how long the sides are drawn or which way it faces. A right angle pointing up and a right angle leaning sideways are both 90°.

Five kinds of angles

Sorted by how far they open, the common angles fall into five types:

Acute: less than 90°, a mouth open just a crack;
Right: exactly 90°, the posture of a wall corner;
Obtuse: more than 90° but less than 180°;
Straight: exactly 180°, the two sides lying flat into a line;
Reflex: more than 180° but less than 360°, like 270° — the angle that swung past flat.

Real examples are everywhere: a pair of scissors half open, a door half ajar. Take this list and go measure a few — angles stop being symbols on paper.

Drag the slider and watch them take shape:

InteractiveAngle Dial

Angle60

Type：Acute= 1.047 rad

180° = π rad

Angles that team up

When several angles share a vertex, two iron rules apply:

Angles laid along a straight line add up to $180°$ ;
Angles sweeping a full circle add up to $360°$ .

From these comes a handy pair: complementary angles add up to 90° ( $30°$ and $60°$ make one), and supplementary angles add up to 180° ( $110°$ and $70°$ make one). Know one angle and the other falls out immediately — the most common free points in geometry problems.

A clock face is a ready-made exercise: the hour hand sweeps 30° every hour, so at 3 o'clock the hands form a 90° angle, and at 4 o'clock a 120° one.

Memory aid

Supplementary = supplies the rest of a straight line, 180°. Complementary = the leftover bit to complete a right angle, 90°. First ask "is the target a line or a corner?", then add or subtract.

Check yourself

Quick quiz

1. What kind of angle is 89°?
2. What is the supplement of 35°?
3. What kind of angle is 270°?