Radians and Degrees
Learn to convert between radians and degrees, and apply these angle measures to calculate arc length and sector area.
Understanding Radians
A radian is an alternative unit for measuring angles. One radian is the angle subtended at the centre of a circle by an arc whose length equals the radius. Since the circumference of a circle is 2πr, a full revolution equals 2π radians.
The key conversion relationship is: π radians = 180°. This means:
Conversion Formulas
Degrees to Radians
radians = degrees × π/180
Radians to Degrees
degrees = radians × 180/π
Common Angle Conversions
| Degrees | Radians | Decimal (approx) |
|---|---|---|
| 0° | 0 | 0 |
| 30° | π/6 | 0.524 |
| 45° | π/4 | 0.785 |
| 60° | π/3 | 1.047 |
| 90° | π/2 | 1.571 |
| 180° | π | 3.142 |
| 270° | 3π/2 | 4.712 |
| 360° | 2π | 6.283 |
Arc Length and Sector Area
When angles are measured in radians, the formulas for arc length and sector area become elegantly simple. For a circle of radius r with a central angle θ (in radians):
Arc Length
l = rθ
where l is the arc length
Sector Area
A = 1/2r²θ
where A is the sector area
Visual: Sector of a Circle
A sector with radius r and central angle θ radians
Why Use Radians?
Radians are the natural unit for angle measurement in mathematics. They simplify many formulas and are essential for calculus. Here are key reasons radians are preferred in advanced mathematics:
Simpler formulas: Arc length l = rθ and sector area A = ½r²θ require no extra conversion factors.
Calculus requires radians: The derivative of sin(x) is cos(x) only when x is in radians.
Small angle approximations: For small θ (in radians), sin θ ≈ θ and tan θ ≈ θ, which are used extensively in physics and engineering.
Key Vocabulary
Radian
The angle subtended at the centre of a circle by an arc equal in length to the radius. One full revolution = 2π radians.
Arc Length
The distance along a curved section of a circle's circumference, calculated as l = rθ.
Sector
A "slice" of a circle bounded by two radii and the arc between them.
Subtended Angle
The angle formed at the centre of a circle by two radii drawn to the endpoints of an arc.
Worked Examples
Convert 150° to radians.
Step 1: Use the formula: radians = degrees × π/180
Step 2: Substitute: 150 × π/180 = 150π/180
Step 3: Simplify by dividing numerator and denominator by 30.
Answer: 150° = 5π/6 radians.
Find the arc length of a sector with radius 10 cm and central angle π/3 radians.
Step 1: Use the formula: l = rθ
Step 2: Substitute: l = 10 × π/3 = 10π/3
Answer: Arc length = 10π/3 ≈ 10.47 cm.
Find the area of a sector with radius 8 cm and central angle π/4 radians.
Step 1: Use the formula: A = 1/2r²θ
Step 2: Substitute: A = 1/2 × 8² × π/4 = 1/2 × 64 × π/4
Answer: Area = 8π ≈ 25.13 cm².
Knowledge Check
Select the correct answer for each question. Click "Check Answer" to see if you are right.
Question 1
Convert 60° to radians.
Question 2
Convert 3π/4 radians to degrees.
Question 3
Find the arc length of a sector with radius 6 cm and angle π/2 radians.
Question 4
Find the area of a sector with radius 10 cm and central angle π/5 radians.
Question 5
How many radians are in a full revolution (360°)?
Key Concepts Summary
- ●π radians = 180°. To convert degrees to radians, multiply by π/180.
- ●To convert radians to degrees, multiply by 180/π.
- ●Arc length formula: l = rθ (angle must be in radians).
- ●Sector area formula: A = ½r²θ (angle must be in radians).
- ●Radians are essential for calculus and many higher-level mathematics applications.