DoAssignment.ca

DoAssignment study guide

TR1 · Convert degree and radian measures and solve arc-length problems

Learn to convert degree and radian measures and solve arc-length problems through clear examples and targeted practice.

Alberta Grade 12 Mathematics

Trigonometry

Why angle measure can describe a length

A radian connects an angle to the length of the arc it cuts off. When an arc has the same length as its circle’s radius, the central angle is one radian. This makes radians especially useful in arc-length problems. Degrees are still valid, but the arc-length formula uses a radian angle, so convert first and always keep track of the unit.

What you will learn

  • Convert angles between degrees and radians.
  • Use arc length, radius, and central angle in one equation.
  • Recognize coterminal angles and check whether an answer fits the context.

Conversion comes from a full turn

One full turn measures 360 degrees or 2π2\pi radians. Divide both measurements by two to get 180 degrees = π\pi radians. For degrees to radians, multiply by π/180\pi/180; in the other direction multiply by 180/π180/\pi. Leave an exact answer as a multiple of π\pi when possible.
For example, 120 degrees becomes 120π/180=2π/3120\pi/180=2\pi/3 radians. A negative angle simply indicates the opposite turning direction. Adding a full turn gives a coterminal angle with the same terminal arm.
\theta_{rad\mathrm{rad}}=\theta_{deg\mathrm{deg}}π180\frac{\pi}{180}, \theta_{deg\mathrm{deg}}=\theta_{rad\mathrm{rad}}180π\frac{180}{\pi}

Arc length needs radians

For a circle of radius r and central angle θ\theta in radians, the subtended arc length is s=rθs=r\theta. The equation is a rearrangement of the radian definition θ=s/r\theta=s/r. Radius and arc length must use the same length unit.
An arc is part of the circumference. For θ=2π\theta=2\pi, the formula gives s=2πrs=2\pi r, the full circumference. This is a useful reasonableness check. If a question gives a diameter, divide it by two before substituting for radius.
s=r\theta (\theta in radians)

Worked example

Find a circular path length

A circular walking path has radius 6 m. What arc length corresponds to a central angle of 75 degrees?
A 75° sector in a circle of radius 6 m
A 75° sector in a circle of radius 6 mThe highlighted arc is 75/360 of the circumference. Its length is 5π/2 m ≈ 7.85 m; the diagram is schematic.θ = 75°θ = 5π/12 radr = 6 ms ≈ 7.85 m

The highlighted arc is 75/360 of the circumference. Its length is 5π/2 m ≈ 7.85 m; the diagram is schematic.

  1. Convert the angle
    Use the degree-to-radian factor before the arc-length formula.
    \theta=75°\frac{\pi}{180°=5π12\frac{5\pi}{12}\ rad\mathrm{rad}
  2. Calculate the arc
    Multiply the radius by the radian measure.
    s=(6\ \mathrm m)(5π12\frac{5\pi}{12})=5π2\frac{5\pi}{2}\ \mathrm m\approx7.85\ \mathrm m
  3. Check the size
    Seventy-five degrees is less than one quarter turn, and 7.85 m is less than a quarter of the full circumference of this circle.
Answer: The arc length is 5π/25\pi/2 m, about 7.85 m.
Check: The full circumference is 12π12\pi m; 75/36075/360 of that is 5π/25\pi/2 m.

Common mistakes and how to avoid them

Substituting a degree measure directly into s=rθs=r\theta.
Correction: Convert the angle to radians first.
Using a given diameter as the radius.
Correction: The radius is half the diameter.

Lesson summary

  • A full turn is 360∘=2π360^\circ=2\pi radians.
  • For arc length, use s=rθs=r\theta with a radian angle.
  • Check units and compare the arc with the circle’s circumference.

Check your understanding

Question 1

A radius is 4 cm and the central angle is π/2\pi/2 radians. What is the arc length?
  1. 2π2\pi cm
  2. 4π4\pi cm
  3. 2 cm
Show answer and explanation
2π2\pi cm
s=rθ=4(π/2)=2πs=r\theta=4(\pi/2)=2\pi cm.

Key terms

Radian
The central angle that subtends an arc equal in length to the radius.
Coterminal angles
Angles whose terminal arms coincide after any whole number of full turns.

Continue through Math 30-1

View the complete Math 30-1 Alberta Grade 12 Mathematics curriculum and lessons

About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Alberta Grade 12 Mathematics (Math 30-1), expectation TR1. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

Official curriculum reference

Report a correction or ask a question