One radian is the angle whose arc is exactly as long as the radius. That definition is what makes the formulas short.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Arc length
Arc length is s = rθ, with θ in radians. In degrees the same formula needs an extra conversion factor.
Sector area
A sector has area ½r²θ, again with θ in radians.
Two kinds of speed
Angular speed measures radians per second. Linear speed measures metres per second. They are linked by v = rω.
Why the outside moves faster
Every point on a spinning wheel shares one angular speed. A larger r then means a larger v, which is why the rim outruns the hub.
What radians were invented for
One radian is the angle subtending an arc equal to the radius. Measuring this way makes arc length simply rθ and sector area ½r²θ, with no fraction of 360 anywhere. That simplification is the entire motivation.
Converting
π radians is 180°, so multiply by π/180 going one way and 180/π going the other. Common angles are worth knowing directly: π/6 is 30°, π/4 is 45°, π/3 is 60°, π/2 is 90°.
Calculus requires radians
The derivative of sin x is cos x only when x is in radians. In degrees an awkward constant appears. Every result in calculus and physics assumes radians, which is why they become the default from here on.
Angular speed
Angular speed is radians per second; linear speed at radius r is rω. A point on the rim of a wheel travels faster than one near the hub while sharing the same angular speed — which is how gears trade speed for torque.
Step 2: Try It Yourself
Tap and try it out.
- Diameter8 cm
- Circumference25.13 cm
- Circumference ÷ diameter3.14
Change the size. The circle gets bigger, but circumference divided by diameter stays at about 3.14 every time. That number is π.
Step 3: Watch an Example
One step at a time.
Watch Ravi Find an Arc
A circle of radius 6 m has a sector of angle 2 radians.
- Step 1
The angle is already in radians, so no conversion is needed.
Step 4: Your Turn
Practice makes it stick.
The Arc
Problem 1 of 2
Radius 5, angle 3 radians. What is the arc length?
The Wheel
Problem 2 of 2
A wheel of radius 0.5 m spins at 4 radians per second. What is the rim speed, in metres per second?
m/s
Around the Rim
1 of 8
Radius 8, angle 2 radians. Arc length?
2 of 8
Radius 4, angle 3 radians. Sector area?
3 of 8
Arc 20, radius 5. What is the angle in radians?
4 of 8
Radius 3 m, angular speed 5 rad/s. Linear speed in m/s?
5 of 8
How many radians is one full turn, to two decimal places?
6 of 8
A wheel turns 10 times per second. Angular speed in radians per second, to one decimal place?
7 of 8
Two points sit on the same spinning wheel, one near the hub and one at the rim. Which statements are true?
8 of 8
Sector area 50, radius 10. Angle in radians?
Step 5: Quick Check
Show what you know.
Question 1 of 2
Radius 7, angle 2 radians. What is the arc length?
Question 2 of 2
Why must θ be in radians for s = rθ?
What You Learned
- Arc length is s = rθ and sector area is ½r²θ, with θ in radians.
- Angular speed is radians per second; linear speed is v = rω.
- Every point on a wheel shares an angular speed but not a linear one.