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Arc Length

Part of the Angles and the Unit Circle section of Coddy's Trigonometry journey. Lesson 18 of 55.

Radians make arcs easy. One radian cuts off an arc one radius long, so θ radians cut off θ radius lengths:

$s = r\cdot \theta$

$A circle of radius 6 with a central angle of π/4 and its arc s$

Radius 6, angle π/4: the arc is s.

With radius 6 and angle $\frac{\pi}{4}$ : $s = \frac{6\cdot \pi}{4}$ , which reduces to $s = \frac{3\pi}{2}$ , about 4.71. The answer stays exact while π stays in it. The angle must be in radians: 6 × 45 is 270, longer than the whole circle.

Challenge

A circle has radius 9. Find the length s of the arc cut off by a central angle of $\frac{2\pi}{3}$ . On the blank board, write s with the numbers put into s = rθ, then a line with s exact, as a multiple of $\pi$ . 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 Is Degrees 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 Recap: Placing Angles

5Arc Length and Sector Area

Arc LengthArcs 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 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