5.1Measuring and Working with Angles
Master degree and radian measure, convert between them, find coterminal and reference angles, and apply the arc length, sector area, and angular speed formulas.
Angle measurement in radians is the foundation of all trigonometry, calculus, and physics. Engineers use arc length and angular speed daily — from designing gears and wheels to programming robotic arms and satellite orbits. Mastering radians now unlocks every trig concept that follows.
Essential Question
How do we measure and work with angles in multiple systems — and why does the radian measure unlock powerful formulas for arc length, sector area, and rotational speed?
Lesson Overview
An angle in standard position has its vertex at the origin and its initial side along the positive x-axis. We measure angles in degrees (a full rotation = 360°) or radians (a full rotation = 2π). To convert, multiply by π/180 (degrees to radians) or 180/π (radians to degrees). Coterminal angles share the same terminal side — add or subtract 360° (or 2π) to generate them. The reference angle is the acute angle between the terminal side and the nearest x-axis. Radian measure is essential for the arc length formula s = rθ, the sector area formula A = ½r²θ, and the relationship between linear speed v and angular speed ω: v = rω.
Conversion Formulas
- Degrees → Radians: multiply by π/180
- Radians → Degrees: multiply by 180/π
- One full rotation: 360° = 2π rad
- Half rotation: 180° = π rad
- Quarter rotation: 90° = π/2 rad
Key Formulas (θ in radians)
- Arc length: s = rθ
- Sector area: A = ½r²θ
- Linear speed: v = rω
- Angular speed: ω = θ/t (radians per unit time)
- Coterminal: θ ± 360° or θ ± 2π
Worked Examples
Convert 135° to radians and convert 5π/6 to degrees.
Degrees → radians: multiply by π/180
135° × (π/180) = 135π/180 = 3π/4
Radians → degrees: multiply by 180/π
(5π/6) × (180/π) = 5 × 30 = 150°
Find two coterminal angles (one positive, one negative) for θ = 110°.
Add 360°: 110° + 360° = 470° (positive coterminal)
Subtract 360°: 110° − 360° = −250° (negative coterminal)
Both 470° and −250° share the same terminal side as 110°.
Find the reference angle for θ = 250°.
250° is in Quadrant III (between 180° and 270°).
Reference angle formula for QIII: θ′ = θ − 180°
θ′ = 250° − 180° = 70°
A circle has radius r = 8 cm. A central angle of θ = 3π/4 radians intercepts an arc. Find (a) the arc length and (b) the area of the sector.
(a) Arc length: s = rθ = 8 × (3π/4) = 6π ≈ 18.85 cm
(b) Sector area: A = ½r²θ = ½ × 64 × (3π/4) = 32 × (3π/4) = 24π ≈ 75.40 cm²
A wheel rotates at an angular speed of ω = 5π/3 radians per second. The wheel has radius 2 feet. Find the linear speed of a point on the rim.
Linear speed formula: v = rω
v = 2 × (5π/3) = 10π/3 feet per second
10π/3 ≈ 10.47 ft/s
Guided Practice
Convert 210° to radians. Leave your answer in terms of π.
Hint: Multiply by π/180 and simplify the fraction. 210/180 = 7/6.
Find two coterminal angles for θ = 7π/4 — one by adding 2π and one by subtracting 2π.
Hint: Add 2π: 7π/4 + 8π/4 = 15π/4. Subtract 2π: 7π/4 − 8π/4 = −π/4.
Find the reference angle for θ = 5π/3 (in radians).
Hint: 5π/3 ≈ 300°, which is in Quadrant IV. For QIV: θ′ = 2π − θ.
A sector has radius r = 5 m and central angle θ = 2 radians. Find the arc length and sector area.
Hint: Use s = rθ and A = ½r²θ. Make sure θ is already in radians — it is!
A bicycle wheel has radius 13 inches and rotates at 3 revolutions per second. Find the angular speed in radians per second and the linear speed of the rim.
Hint: Each revolution = 2π radians. Angular speed ω = 3 × 2π. Then v = rω.
Key Vocabulary
Standard Position
An angle is in standard position when its vertex is at the origin and its initial side lies along the positive x-axis. The terminal side is where the rotation ends.
Radian
The angle subtended at the center of a circle by an arc equal in length to the radius. One full rotation = 2π radians. Radians are dimensionless and required for arc/area formulas.
Degree
A unit of angle measure where one full rotation equals 360°. Degrees are convenient for navigation and geometry but must be converted to radians for calculus and trig formulas.
Coterminal Angles
Two angles in standard position that share the same terminal side. Generated by adding or subtracting multiples of 360° (or 2π radians).
Reference Angle
The positive acute angle formed between the terminal side of an angle and the nearest x-axis. Used to evaluate trig functions for any angle using the values from Quadrant I.
Arc Length
The distance along the curved part of a circle intercepted by a central angle. Formula: s = rθ, where θ must be in radians.
Sector Area
The area of the 'pie slice' region bounded by two radii and an arc. Formula: A = ½r²θ, where θ must be in radians.
Angular Speed (ω)
The rate at which an angle changes over time, measured in radians per unit time. Related to linear speed by v = rω.
Check Your Understanding
Interactive Practice — 5 Questions
Convert 240° to radians.
Which angle is coterminal with 50°?
What is the reference angle for θ = 7π/6?
A circle has r = 10 and central angle θ = π/5 radians. What is the arc length?
A wheel of radius 3 ft spins at ω = 4 rad/s. What is the linear speed of the rim?
Common Mistakes
Using the arc length formula s = rθ with θ in degrees.
θ must be in radians for s = rθ and A = ½r²θ. Always convert degrees to radians first.
Confusing the reference angle with the coterminal angle.
A reference angle is always acute (between 0° and 90°) and is measured from the x-axis. Coterminal angles have the same terminal side but differ by full rotations.
Finding the reference angle for a QII angle as θ − 180° instead of 180° − θ.
In QII, the reference angle is 180° − θ (or π − θ). In QIII it is θ − 180°. In QIV it is 360° − θ.
Multiplying by 180 instead of π/180 when converting degrees to radians.
To go degrees → radians, multiply by π/180. To go radians → degrees, multiply by 180/π. The π goes with radians.
Math Tips
Memorize the "big five": 30°=π/6, 45°=π/4, 60°=π/3, 90°=π/2, 180°=π. All others follow from multiples.
Quick coterminal check: keep adding or subtracting 360° (or 2π) until the angle is in [0°, 360°) or [0, 2π). That is the principal angle.
Reference angles are always positive and always between 0° and 90°. If you get a reference angle greater than 90°, recheck your quadrant.
Angular speed ω is in radians per unit time. If a problem gives revolutions per minute (rpm), convert: 1 revolution = 2π radians.
The sector area formula A = ½r²θ looks like the triangle area formula ½bh — that is not a coincidence. As θ → 0, the sector approaches a triangle.