Arc Length
The radian definition was built from the relationship between an arc and its radius. Now we formalize that relationship into the arc length formula — one of the most elegant equations in all of geometry.
Introduction
In Lesson 3.1, we defined a radian as . Today we rearrange that definition into a formula that directly computes the length of any circular arc, given only the radius and the central angle.
Past Knowledge
A radian is defined as . You can convert between degrees and radians.
Today's Goal
Calculate the arc length using where must be in radians.
Future Success
Arc length is fundamental to calculus, circular motion in physics, and designing curved structures in engineering.
Key Concepts
The Arc Length Formula
Starting from the radian definition , multiply both sides by :
Arc Length Formula
where = arc length, = radius, = central angle in radians
Why Must θ Be in Radians?
The formula was derived from the radian definition. It only works because radians are a pure ratio (arc ÷ radius). If you plug in degrees, the formula breaks — you would need an extra factor of to compensate.
Three Variations
The formula can be rearranged to solve for any of the three variables:
| Solve For | Formula |
|---|---|
| Arc length | |
| Radius | |
| Angle |
Worked Examples
Finding Arc Length
Question: A circle has radius cm. Find the arc length intercepted by a central angle of radians.
Step 1: Apply the formula.
Step 2: Simplify.
Final Answer: cm
Angle Given in Degrees (Conversion Required)
Question: A pizza has radius inches. Find the length of the crust along a slice.
Step 1: Convert to radians.
Step 2: Apply the arc length formula.
Final Answer: The crust is inches long.
Solving for Radius
Question: A central angle of radians cuts off an arc meters long. Find the radius.
Step 1: Use the rearranged formula.
Step 2: Divide by multiplying by the reciprocal.
Final Answer: meters
Common Pitfalls
Plugging in Degrees Directly
The arc length formula requires in radians. Using when the angle is is wildly incorrect. You must convert to first.
❌ cm
✅ cm
Real-Life Applications
Windshield Wipers
Automotive engineers use to calculate how much glass area a windshield wiper covers per sweep. The wiper blade is the radius, and the sweep angle (typically about radians, or ) determines the arc length at the blade tip. Knowing these values ensures the blade clears enough area for safe visibility.
Running Track Curves
Outdoor running tracks have semicircular ends. Each lane has a different radius, so the arc length varies by lane. That's why sprinters in outer lanes start ahead — the staggered start compensates for the longer arc length runners in outer lanes must travel around the curve.
Practice Quiz
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