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Convert angle measure and use the radian arc relation

Start a set of 8
Step 5 of 5
Radius
\(\displaystyle 9\text{ cm}\)
Minor central angle
\(\displaystyle 60^\circ\)

Which calculation gives the minor arc length?

Year 11, VCE Units 1 & 2, Chapter 9, 4 difficulty levels, Calculator allowedBuild a worksheet

Circular measure and arc length

Relate arc length to radius, convert angle units and choose the minor or major arc

\[l=r\theta,\quad 180^{\circ}=\pi\ \mathrm{rad}\]
  1. Understand a radian One radian is the central angle subtended by an arc equal in length to the radius. Divide arc length by radius to measure an angle in radians \(\displaystyle \theta=\frac{l}{r},\qquad l=r\Longrightarrow\theta=1\text{ rad}\)
  2. Convert angle units A full turn has the same size in either unit. Multiply by the appropriate conversion factor \(\displaystyle \begin{gathered}360^\circ=2\pi\text{ rad}\\\theta_{\rm rad}=\theta_{\rm deg}\frac{\pi}{180}\\\theta_{\rm deg}=\theta_{\rm rad}\frac{180}{\pi}\end{gathered}\)
  3. Use radians in the arc relation Use the radius and arc length in the same length unit. The angle must be in radians \(\displaystyle l=r\theta\)
  4. Distinguish the two arcs The minor arc is shorter than a semicircle. The major arc is longer than a semicircle and completes the full turn \(\displaystyle \begin{gathered}0<\theta_{\rm minor}<\pi\\\theta_{\rm major}=2\pi-\theta_{\rm minor}\end{gathered}\)

Worked example

A circle has radius nine centimetres. Its minor central angle is sixty degrees. Find the exact length of the corresponding major arc

Circle with centre O and radius nine centimetres. Radii OA and OB enclose the sixty-degree minor angle. Dashed minor arc AB and solid major arc AB complete the circle9 cmOAB
Dashed: minor arc AB; solid: major arc AB. The stated angle is the minor angle between the radii
\[r=9\text{ cm},\qquad\theta_{\rm minor}=60^\circ\]
  1. \(\displaystyle 60^{\circ}\times\frac{\pi}{180^{\circ}}=\frac{\pi}{3}\ \mathrm{rad}\)Convert the degree measure to radians before using the arc relation
  2. \(\displaystyle \theta_{\rm major}=2\pi-\frac{\pi}{3}=\frac{5\pi}{3}\)Subtract the minor angle from one full turn to get the major-arc angle
  3. \(\displaystyle l=r\theta=9\times\frac{5\pi}{3}=15\pi\ \mathrm{cm}\)Substitute the radius and the major-arc angle in radians

Common mistakes

Using degrees in the arc formula
✕\(\displaystyle l=9\left(60\right)=540\ \mathrm{cm}\)✓\(\displaystyle 60^{\circ}=\frac{\pi}{3}\ \mathrm{rad},\quad l=9\left(\frac{\pi}{3}\right)=3\pi\ \mathrm{cm}\)
The arc-length relation requires radians, so convert the degree measure before substitution
Forgetting the major-arc complement
✕\(\displaystyle \theta_{\rm major}=\theta_{\rm minor},\quad l=r\theta_{\rm minor}\)✓\(\displaystyle \theta_{\rm major}=2\pi-\theta_{\rm minor},\quad l=r\left(2\pi-\theta_{\rm minor}\right)\)
A major arc is the rest of the full turn, so its central angle is the full turn minus the minor central angle
Reversing the conversion factor
✕\(\displaystyle 45^{\circ}\times\frac{180^{\circ}}{\pi}\)✓\(\displaystyle 45^{\circ}\times\frac{\pi}{180^{\circ}}\)
Degrees to radians uses π divided by 180 degrees, while 180 degrees divided by π changes radians to degrees

Difficulty levels

introConvert one exact angle between degrees and π-radians
standardUse an angle already in radians in the forward arc-length relation
challengeConvert before calculating, use a major arc or solve the arc relation in reverse
examChoose the correct unit decision and then calculate one arc-length value

Questions about this topic

Why must the arc formula use radians?+

Radian measure is arc length divided by radius. Rearranging this definition gives the arc formula

\[\theta=\frac{l}{r}\Longrightarrow l=r\theta\]
How is the major-arc angle found?+

Subtract the given minor central angle from one full turn, then use radians in the arc formula

\[\theta_{\rm major}=2\pi-\theta_{\rm minor}\]
What is an exact arc length?+

Keep pi exact instead of replacing it with a rounded decimal

\[15\pi\text{ cm}\]
Can I use a calculator?+

A calculator is allowed unless a question is marked no calculator

Which year level and chapter is this?+

Year 11, VCE Units 1 & 2, Mathematical Methods, chapter 9: Circular functions

In this chapter

Convert angle measure and use the radian arc relationLocate extended angles and generate coterminal anglesDeduce exact trigonometric ratios from one ratioUse unit-circle coordinates to evaluate circular functionsEvaluate exact circular-function values by symmetry and periodicityKey points and features of the basic sine and cosine graphsLocate and bisect a circular-equation rootUse local polynomial approximations for circular functions