4.1: Radians and Arc Length | Precalculus

How it works

A radian measures an angle by the ratio of arc length to radius. Use radians in arc-length and sector-area formulas. Convert degrees before substituting, and distinguish arc length from the straight chord length.

s=r\theta,\quad A=\tfrac12r^2\theta\quad(\theta\text{ in radians})

Worked examples

Example 1

Convert $\displaystyle 150^\circ$ to radians.

$\displaystyle 150\pi/180=5\pi/6$ radians.

Example 2

A radius- $\displaystyle 4$ circle has central angle $\displaystyle \pi/3$ . Find arc length.

$\displaystyle s=4(\pi/3)=4\pi/3$ .

Problems

Problem 4.1.1

A radius- $\displaystyle 6$ circle has a $\displaystyle 60^\circ$ central angle. Find arc length and sector area.

Sector of a circle with radius 6 and central angle 60 degrees. — Convert 60 degrees to pi/3 radians before using the formulas.

Show worked solution

Convert to $\displaystyle \pi/3$ . Then $\displaystyle s=6(\pi/3)=2\pi$ , and $\displaystyle A=(1/2)(36)(\pi/3)=6\pi$ .

Sector of a circle with radius 6 and central angle 60 degrees. The solution adds labeled relationships. — Arc length is 2 pi, and sector area is 6 pi.

Problem 4.1.2

Convert $\displaystyle 7\pi/4$ radians to degrees.

Show worked solution

$\displaystyle 7\pi\cdot180/(4\pi)=315^\circ$ .

Problem 4.1.3

An arc has length $\displaystyle 15$ on a radius- $\displaystyle 5$ circle. Find its central angle in radians.

Show worked solution

$\displaystyle \theta=s/r=15/5=3$ radians.

Lesson Problems PDF Solutions PDF Unit MCQ

How it works

Worked examples

Example 1

Example 2

Problems

Problem 4.1.1

Problem 4.1.2

Problem 4.1.3

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