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What a Radian Is

Part of the Angles and the Unit Circle section of Coddy's Trigonometry journey. Lesson 14 of 57.

Take a piece of string as long as the radius and lay it along the circle. The angle at the centre that this arc makes is one radian, about 57.3°. It is a unit the circle chooses for itself: it does not depend on how big the circle is.

$An arc as long as the radius r, making an angle of one radian at the centre$

Arc = radius: the angle is 1 radian.

The whole circle is 2π×r long, so it holds 2π radius lengths: a full turn is 2π radians and a half turn is π. Everything else is a fraction of those. π/2 is a quarter turn, the positive y-axis. 3π/4 is three quarters of a half turn, so it ends in quadrant II, 45° short of the negative x-axis.

Challenge

The circle is labelled in radians. Drag the point to 2π/3, then press Check.

Test yourself

This lesson includes a short quiz. Start the lesson to answer it and track your progress.

Test yourself

This lesson includes a short quiz. Start the lesson to answer it and track your progress.

All lessons in Angles and the Unit Circle

1A Point on a Circle

Height on the Circle Across: the Cosine Same Height, Two Angles Recap: Height and Across

4Radians

What a Radian IsDegrees to Radians Radians to Degrees Radians Without π Turns in Radians Recap: Radians

7Reference Angles and Signs

Reference Angles Reference in III and IV Signs by Quadrant Sign and Size Signs at Any Angle Recap: Reference Angles

10Six Functions, Any Circle

Any Circle, Any Radius From a Point to the Angle Secant, Cosecant, Cotangent Reciprocals in Radians Recap: Six Functions

2Standard Position

Angles in Standard Position Negative Angles Quadrants and Axes Recap: Placing Angles

5Arc Length and Sector Area

Arc Length Arcs From Degrees Finding the Angle Sector Area Recap: A Slice of Pizza

8All Four Quadrants

Points in Quadrant II Quadrants III and IV Every Angle for a Value Points in Radians Exact Values in Radians Recap: Four Quadrants

11Final Challenges

The Minute Hand The Ferris Wheel Seat Four Angles, Two Turns The Garden Sprinkler A Negative Radian Point Which Angle?

3Coterminal Angles

Coterminal Angles Back Into One Turn Values Repeat Every Turn Every Turn at Once Recap: Coterminal Angles

6The Unit Circle

Why (cos θ, sin θ)The 30° and 60° Points The 45° Point Points Off the Table Recap: Quadrant I Points

9Tangent on the Circle

Tangent Is y over x Where Tangent Is Undefined Tangent in Radians Every Angle for a Tangent Recap: Tangent