Arc length and sector area use the same fraction of a circle, but they scale different whole-circle quantities. Arc length takes a fraction of circumference; sector area takes a fraction of area.
Whole-circle formulas
Circumference is a length, so its units are linear.
Area uses square units.
One fraction connects angle, arc, and sector
| Quantity | Whole | Part for θ degrees |
|---|---|---|
| Angle | \(360^\circ\) | \(\theta/360\) |
| Arc length | \(2\pi r\) | \(L=\frac{\theta}{360}(2\pi r)\) |
| Sector area | \(\pi r^2\) | \(A_s=\frac{\theta}{360}(\pi r^2)\) |
Find two related quantities
A circle has radius 12 and central angle \(150^\circ\). Find the exact arc length and sector area.
- Reduce the fraction
\(150/360=5/12\).
- Arc
\(L=(5/12)(24\pi)=10\pi\).
- Sector
\(A_s=(5/12)(144\pi)=60\pi\).
Solve backward for radius or angle
Reverse workflow
- Choose the whole
Use \(2\pi r\) for an arc or \(\pi r^2\) for a sector.
- Write the fraction
Use \(\theta/360\), then solve the resulting equation.
- Check scale
A fraction below one must produce a part smaller than the whole.
Wheel revolutions and unit conversion
n is the number of complete revolutions, assuming no slipping.
Convert before dividing
A wheel has radius 0.35 meter. Approximately how many revolutions carry it 220 meters?
- One revolution
\(C=2\pi(0.35)=0.7\pi\) meters.
- Divide distance
\(n=220/(0.7\pi)\approx100\).
Fraction check
A \(72^\circ\) sector has radius 10. Find its arc length and area.
Show answer and explanation
Answer: \(4\pi\) units and \(20\pi\) square units.
The sector is one fifth of the circle.
Key takeaways
- Central-angle fraction equals arc-length fraction equals sector-area fraction.
- Keep exact pi until a decimal is requested.
- Convert all wheel distances to one unit before dividing by circumference.
- Radians use a 2π full turn, not 360 degrees.
Put these notes into practice
Apply the ideas with SAT-style questions, then reinforce key details with flashcards.