Radians, the circle's own unit

6 minAS Level1.4.1

AI-drafted, independently checked · not yet teacher-reviewed

By the end of this lesson

By the end you can say exactly what one radian is, convert any angle between degrees and radians, and rattle off the key angles like $\tfrac{\pi}{6} = 30^\circ$ without thinking.

Why 360 anyway?

Nobody knows for sure why a full turn is $360^\circ$ . The best guess: ancient Babylonians liked numbers that divide nicely, and a year is roughly 360 days.

It's a human choice, not a maths one.

Radians are the circle's own unit. Instead of chopping a turn into 360 arbitrary bits, you ask: how many radius-lengths of arc does this angle cut off?

Definition

Radian

One radian (1 rad) is the angle subtended at the centre of a circle by an arc whose length is equal to the radius of the circle.

In plain words:

An arc is a piece of the circle's edge.
"The angle subtended at the centre" means the angle you get by drawing lines from both ends of the arc to the centre.
So: take an arc exactly one radius long. The angle it makes at the centre is 1 radian.

Wrap the radius around the edge of the circle. The angle it reaches at the centre is one radian, about

57.3^\circ

Where π comes from

The whole circumference (the distance all the way round) is $2\pi r$ .

That's $2\pi$ radius-lengths of arc. So a full turn is $2\pi$ radians, about 6.28.

$2\pi \text{ rad} = 360^\circ$

Halve both sides:

$\pi \text{ rad} = 180^\circ$

That one line is the whole conversion machine. Everything else is multiplying.

Quick check

Multiple choiceQuestion 1 · Recall

Fill in the blanksQuestion 2 · Recall

Answer the 2 questions above to continue.