Math101 · Independent Ontario learning library
Math101learn.math101.caSector Area
A sector is the region bounded by two radii and their intercepted arc.
Sector area measures portions of circular fields, pie charts, rotating sweeps, and mechanical components. It reinforces proportional reasoning and radians.
Intuition and core definition
A sector is the region bounded by two radii and their intercepted arc. Its area is the same fraction of full circle area as its central angle is of a full turn: $A_s=(\theta/360^\circ)\pi r^2$, or $A_s=\frac12r^2\theta$ when $\theta$ is in radians.
Notation, language, and conditions
$r>0$ and $\theta$ must be a central angle. Degree and radian formulas are not interchangeable without conversion. A minor sector corresponds to an angle below $180^\circ$; a major sector uses the remaining angle $360^\circ-\theta$.
Why this idea matters
Sector area scales the full circle's area by the central angle's fraction of one revolution.
A dependable method
- Identify radius, central angle, and angle units.
- Choose the degree-fraction or radian formula.
- Calculate the turn fraction and multiply by $\pi r^2$.
- Keep an exact $\pi$ form before rounding.
- Use square units and compare with the entire disk area.
Worked example
Representations and interpretation
A sector is a wedge-shaped fraction of a disk. The proportions $A_s/(\pi r^2)=\theta/360^\circ$ and $s/(2\pi r)=\theta/360^\circ$ align area and arc length for the same angle.
Reasoning about variations
Doubling radius with fixed angle quadruples sector area. Two sectors with equal arc length but different radii need not have equal area because their angles differ.
Common mistakes
How to check your work
- Ensure the sector area lies between zero and full area.
- Compare sector/full-area ratio with angle/full-turn ratio.
- Add minor and major sectors to recover $\pi r^2$.
Practice
- Find the area of a $90^\circ$ sector with radius $8$.
- Find sector area for $r=3$ and $\theta=2$ radians.
- If angle stays fixed and radius triples, by what factor does sector area change?
Answers and brief solutions
Show answers
- $16\pi$ A quarter of $64\pi$ is $16\pi$.
- $9$ $\frac12(3)^2(2)=9$.
- $9$ Sector area depends on $r^2$.
Synthesis and transfer
A rotating sprinkler waters a sector; comparing its angle with a full turn predicts the fraction of lawn covered, and square units distinguish area from arc length.
A sprinkler covering radius $10$ m through $72^\circ$ waters one fifth of a disk, so the area is $20\pi$ square metres. The corresponding arc is one fifth of the circumference, or $4\pi$ metres; keeping units visible prevents those two quantities from being swapped. In radians, the same area is $\frac12r^2\theta$, with $72^\circ=2\pi/5$. Doubling the angle doubles the sector area, but doubling radius multiplies it by four. A sector larger than a semicircle is valid as long as the chosen central angle describes the intended major region.
Related topics
Teaching and accessibility note
Try it yourself
Hints are part of learning. Open one whenever it makes the next step feel possible.
Find the area of a $90^\circ$ sector with radius $8$.
- A quarter of $64\pi$ is $16\pi$.
End of lesson
Nice work making it this far.
Understanding grows through return visits. Save this lesson, try the practice, or continue when you are ready.
