Perimeter of a sector
The perimeter of a sector is the curved arc together with its two straight radii. For a major sector, use the remaining angle before finding the arc
\[P=2r+\frac{\theta}{360}\times2\pi r\]
- Use the required angle For a major sector, subtract the stated minor angle from 360 degrees \(\displaystyle \theta_{\text{used}}=360^\circ-120^\circ=240^\circ\)
- Find the circle fraction Divide the angle used by 360 degrees \(\displaystyle \frac{240}{360}=\frac{2}{3}\)
- Find the arc Multiply the circle fraction by the full circumference \(\displaystyle s=\frac{240}{360}\times2\pi(6)=8\pi\text{ cm}\)
- Add the radii Add two radii to the arc and round only at the end when a decimal answer is requested \(\displaystyle P=2(6)+8\pi=12+8\pi\text{ cm}\)
Worked example
A minor sector has radius 9 cm and central angle 80°. Find its perimeter to 2 decimal places
\[r=9\text{ cm},\quad\theta=80^\circ\]
- \(\displaystyle \frac{\theta}{360}=\frac{80}{360}=\frac{2}{9}\)The minor sector is 2/9 of the full circle
- \(\displaystyle s=\frac{2}{9}\times2\pi(9)=4\pi\text{ cm}\)Use the circle fraction to find the curved arc
- \(\displaystyle \begin{gathered}P=2(9)+4\pi\\P=(18+4\pi)\text{ cm}\end{gathered}\)Add the two radii to the exact arc length
- \(\displaystyle P\approx30.57\text{ cm}\)Round the exact perimeter once to 2 decimal places
Common mistakes
Using the minor angle for a major sector
✕\(\displaystyle \begin{gathered}P=2(6)+\frac{120}{360}\times2\pi(6)\\P=12+4\pi\end{gathered}\)✓\(\displaystyle \begin{gathered}P=2(6)+\frac{240}{360}\times2\pi(6)\\P=12+8\pi\end{gathered}\)
A major sector uses the remaining angle, 360 degrees minus the stated minor angleLeaving out the two radii
✕\(\displaystyle P=8\pi\text{ cm}\)✓\(\displaystyle P=12+8\pi\text{ cm}\)
The perimeter follows the arc and both straight radii, not the arc aloneUsing the full circumference
✕\(\displaystyle P=2(6)+2\pi(6)=12+12\pi\)✓\(\displaystyle \begin{gathered}P=2(6)+\frac{240}{360}\times2\pi(6)\\P=12+8\pi\end{gathered}\)
The arc is only the angle fraction of the full circumferenceDifficulty levels
introFind an exact perimeter for a semicircle or quadrant when the radius is given
standardFind a minor-sector perimeter to a stated number of decimal places
challengeHandle a major sector, a stated diameter, and either an exact or rounded perimeter
examChoose the rounded major-sector perimeter from options containing common arc and radius errors
Questions about this topic
What lengths make up a sector perimeter?+
A sector perimeter contains the curved arc and the two radii that bound the arc
How is the angle for a major sector found?+
Subtract the stated minor angle from 360 degrees because the minor and major sectors make one full turn
Why is the arc not the whole circumference?+
The arc covers only the fraction of the circle represented by the sector angle, so use angle divided by 360.
Can I use a calculator?+
A calculator is allowed unless a question is marked no calculator
Which year level and chapter is this?+
Year 9 Mathematics, chapter 5: Measurement
In this chapter
5A Length conversion and perimeter: Convert measurement units5A Length conversion and perimeter: Perimeter and area of standard shapesFind an unknown side from a perimeter5A Length conversion and perimeter: Solve a linear equation5A Length conversion and perimeter: Substitute values into a formula5A Length conversion and perimeter: Round measurements to decimal places5B Circumference, sectors and exact π: Circumference and area of a circle5B Circumference, sectors and exact π: Arc length and sector areaPerimeter of a sector5B Circumference, sectors and exact π: Simplify exact expressions involving pi5B Circumference, sectors and exact π: Substitute values into a formula5B Circumference, sectors and exact π: Round measurements to decimal places5C Area formulas and units: Convert measurement units5C Area formulas and units: Perimeter and area of standard shapes5C Area formulas and units: Circumference and area of a circle5C Area formulas and units: Arc length and sector area5C Area formulas and units: Simplify exact expressions involving pi5C Area formulas and units: Substitute values into a formula5C Area formulas and units: Round measurements to decimal placesPerimeter and area of composite shapes5D Composite perimeter and area: Find a side of a right-angled triangle5D Composite perimeter and area: Substitute values into a formula5D Composite perimeter and area: Round measurements to decimal places5E Right-prism surface area: Surface area of cuboidsSurface area of a triangular prism5E Right-prism surface area: Find a side of a right-angled triangle5E Right-prism surface area: Substitute values into a formula5E Right-prism surface area: Round measurements to decimal placesPyramid surface area: Surface area of square pyramids (optional extension)5F Cylinder surface area: Surface area of cylindersSurface area of a cylinder portion5F Cylinder surface area: Substitute values into a formula5F Cylinder surface area: Round measurements to decimal places5G Volume and capacity: Convert measurement units5G Volume and capacity: Volume of right prisms5G Volume and capacity: Capacity of a container5G Volume and capacity: Solve a linear equation5G Volume and capacity: Substitute values into a formula5G Volume and capacity: Round measurements to decimal places5H Cylinder volume and capacity: Volume of cylinders5H Cylinder volume and capacity: Capacity of cylindersVolume of a cylinder portion5H Cylinder volume and capacity: Substitute values into a formula5H Cylinder volume and capacity: Round measurements to decimal placesSmallest unit and limits of accuracy