Angles can be measured in degrees or radians, and both describe the same amount of rotation. Degrees divide a full circle into 360 equal parts, which is useful for navigation, geometry, and everyday angle descriptions. Radians measure angle using the relationship between a circle's radius and the arc length cut off by the angle.
Understanding both units helps connect geometry, trigonometry, and motion around circles.
A radian is defined so that an angle of 1 radian cuts off an arc length equal to the radius of the circle. Since the circumference of a circle is 2πr, a full rotation is 2π radians, which equals 360 degrees. This makes π radians equal to 180 degrees, giving the main conversion link between the two systems.
Radians are especially natural in formulas such as arc length s = rθ and angular speed because they come directly from circle geometry.
Key Facts
- Full circle: 360° = 2π radians
- Half circle: 180° = π radians
- Degree to radian conversion: radians = degrees × π/180
- Radian to degree conversion: degrees = radians × 180/π
- Arc length formula: s = rθ, where θ is in radians
- Sector area formula: A = 1/2 r^2θ, where θ is in radians
Vocabulary
- Degree
- A degree is an angle unit equal to 1/360 of a full rotation.
- Radian
- A radian is an angle unit where the arc length equals the radius of the circle.
- Arc length
- Arc length is the distance along the curved edge of a circle between two points.
- Central angle
- A central angle is an angle whose vertex is at the center of a circle.
- Sector
- A sector is the wedge-shaped region of a circle enclosed by two radii and an arc.
Common Mistakes to Avoid
- Using degree values directly in s = rθ is wrong because the formula requires θ in radians.
- Forgetting that π radians equals 180° leads to incorrect conversions, so always start from π rad = 180°.
- Dividing by π when converting degrees to radians is wrong because degrees should be multiplied by π/180.
- Treating radians as a separate kind of angle is misleading because radians and degrees measure the same rotation in different units.
Practice Questions
- 1 Convert 135° to radians in exact form.
- 2 A circle has radius 8 cm and a central angle of π/3 radians. Find the arc length.
- 3 Explain why radians make the formula s = rθ simpler than using degrees.