Area of a sector
A sector area is its central-angle fraction of the circle area. Use the required minor or major angle in degrees
\[A=\frac{\theta}{360}\pi r^2\]
- Find the circle fraction Divide the central angle by 360 degrees to find the fraction of a full turn \(\displaystyle \begin{gathered}\theta=120^\circ\\\frac{\theta}{360}=\frac{120}{360}=\frac{1}{3}\end{gathered}\)
- Use the sector-area formula Sector area is the same fraction of the circle area and contains r² \(\displaystyle \begin{gathered}A=\frac{120}{360}\times\pi(6)^2\\A=12\pi\text{ cm}^2\end{gathered}\)
- Find the major angle For a major piece, subtract the stated minor angle from 360 degrees before applying the formula \(\displaystyle \begin{gathered}\theta_{\mathrm{major}}=360^\circ-80^\circ\\\theta_{\mathrm{major}}=280^\circ\end{gathered}\)
Worked example
A minor sector has radius 9 cm and central angle 80°. Find its area to 2 decimal places
\[r=9\text{ cm},\quad\theta=80^\circ\]
- \(\displaystyle \frac{\theta}{360}=\frac{80}{360}=\frac{2}{9}\)The sector occupies two ninths of the full circle
- \(\displaystyle A=\frac{2}{9}\pi(9)^2=18\pi\text{ cm}^2\)Use the sector-area formula with the radius squared
- \(\displaystyle 18\pi\text{ cm}^2\approx56.55\text{ cm}^2\)Round the exact area once at the end to 2 decimal places
Common mistakes
Ignoring the angle fraction
✕\(\displaystyle A=\pi(4)^2=16\pi\text{ cm}^2\)✓\(\displaystyle \begin{gathered}A=\frac{45}{360}\times\pi(4)^2\\A=2\pi\text{ cm}^2\end{gathered}\)
A sector covers only the central angle's share of the circle, so the full-circle area is too largeSwapping arc and sector formulas
✕\(\displaystyle \begin{gathered}A=\frac{120}{360}\times2\pi(6)\\A=4\pi\text{ cm}^2\end{gathered}\)✓\(\displaystyle \begin{gathered}A=\frac{120}{360}\times\pi(6)^2\\A=12\pi\text{ cm}^2\end{gathered}\)
Sector area uses the radius squared, while circumference uses the radius to give a lengthUsing the minor angle for a major sector
✕\(\displaystyle A=\frac{80}{360}\pi(5)^2\)✓\(\displaystyle \begin{gathered}A=\frac{360-80}{360}\pi(5)^2\\A=\frac{280}{360}\pi(5)^2\end{gathered}\)
A major sector uses the remaining angle, 360 degrees minus the stated minor angleDifficulty levels
standardFind a minor sector area from its radius
challengeUse a diameter and the remaining angle for major sector area, exactly or rounded
examRecognise the correct major sector area from distinct options
Questions about this topic
Why is the angle divided by 360?+
A full turn is 360 degrees, so divide the central angle by 360 to find the fraction of the circle it covers
How are arc length and sector area different?+
Arc length measures the curved boundary and uses one radius. Sector area measures the region and uses the radius squared
How is a major sector angle found?+
If the minor angle is θ, the major angle is 360° − θ because the two pieces make one full turn
Can I use a calculator?+
A calculator is allowed unless a question is marked no calculator
Which year level and chapter is this?+
Year 10 Mathematics, chapter 1: Measurement
In this chapter
Convert measurement unitsPerimeter of standard shapesFind an unknown side from a perimeterCircumference of a circlePerimeter of composite shapesArea of standard shapesArea of a circleArea of a sectorArea of composite shapes1F Prism surface area: Surface area of solidsSurface area of a triangular prismExposed surface area of joined cuboids1G Cylinder surface area: Surface area of solidsSurface area of a cylinder portionSurface area of a cylinder joined to a cuboidVolume of prisms and cylindersCapacity of a containerVolume of a cylinder portionJoined and hollow prism-cylinder volumesModel rainwater collection and storageSmallest unit and limits of accuracyMeasurement bounds in calculationsVolume of cones, pyramids and spheres (optional extension)1J Level 10A extension: pyramids, cones and spheres: Surface area of solids (optional extension)Surface areas of advanced solids and composites (optional extension)