Arc length and sector area describe parts of a circle using the radius and the central angle. They are important because many real objects, such as wheels, gears, pizza slices, radar sweeps, and circular tracks, involve only part of a full circle. The cleanest formulas use radians, because radians connect angle measure directly to distance around a circle.
When the angle is in radians, the equations become short and easy to apply.
A sector is the region inside a circle bounded by two radii and the arc between them. The arc length grows in direct proportion to both the radius and the central angle, while the sector area grows with the square of the radius and the angle. This means doubling the angle doubles both arc length and sector area, but doubling the radius doubles arc length and multiplies sector area by four.
Worked examples usually start by converting degrees to radians, then substituting into s = rθ or A = 1/2 r²θ.
Key Facts
- Arc length formula in radians: s = rθ
- Sector area formula in radians: A = 1/2 r²θ
- A full circle has angle 2π radians, circumference C = 2πr, and area A = πr²
- Degree to radian conversion: θ radians = θ degrees × π/180
- Fraction of a circle: θ/(2π) when θ is measured in radians
- Sector area can also be found by A = (θ/360)πr² when θ is measured in degrees
Vocabulary
- Arc length
- Arc length is the distance along a curved part of a circle.
- Sector
- A sector is the region of a circle enclosed by two radii and the arc between them.
- Central angle
- A central angle is an angle whose vertex is at the center of a circle.
- Radian
- A radian is an angle measure where one radian subtends an arc length equal to the radius.
- Radius
- The radius is the distance from the center of a circle to any point on the circle.
Common Mistakes to Avoid
- Using degrees directly in s = rθ, which is wrong because this formula requires θ in radians. Convert degrees to radians first or use a degree-based fraction formula.
- Forgetting to square the radius in A = 1/2 r²θ, which gives an area with the wrong size and units. Area must be measured in square units.
- Confusing arc length with sector area, which mixes a boundary distance with a two-dimensional region. Arc length uses linear units, while sector area uses square units.
- Using the diameter instead of the radius, which makes the answer too large. The formulas s = rθ and A = 1/2 r²θ both require radius.
Practice Questions
- 1 A circle has radius 8 cm and central angle π/3 radians. Find the arc length and the sector area.
- 2 A sector has radius 10 m and central angle 72 degrees. Convert the angle to radians, then find the arc length and sector area.
- 3 Two sectors have the same central angle, but one circle has twice the radius of the other. Explain how their arc lengths and sector areas compare.