Use the central-angle fraction of a complete circle.
Core & Extended · Syllabus C5.3, E5.3
Notes and key methods
A sector with central angle θ degrees occupies \(\displaystyle \frac{\theta }{360}\) of a circle. Its arc length is that fraction of 2πr, and its area is the same fraction of \(\displaystyle \pi r^{2}\). Core examples use angles that divide a full turn exactly.
A sector perimeter includes two straight radii as well as the arc. The arc is a length; the sector is an area. Do not halve the sector formula again, and keep the central angle distinct from an angle at the circumference.
Check the conditions
Use the stated units and precision.
Show the method as well as the final answer.
Three worked examples
Example 1 · Simple sectors and arc fractions
A sector has radius 4 cm and central angle \(120\)°.
Find the fraction of a complete circle occupied by the sector.
Write its arc length as kπ cm. Find k.
Write its area as \(\displaystyle k\pi \mathrm{cm}^{2}\). Find k.
Its perimeter is aπ+b cm. Find b.
Show the reasoning
(a)
Express the central angle as a fraction of a full turn.
Divide \(120\) by 360 to obtain \(\displaystyle \frac{1}{3}\).
Answer: \(\displaystyle \frac{1}{3}\).
(b)
Apply the angle fraction to circumference.
The circle circumference coefficient 8 is multiplied by \(\displaystyle \frac{1}{3}\).
Answer: \(\displaystyle \frac{8}{3}\).
(c)
Apply the angle fraction to circle area.
The circle area coefficient 16 is multiplied by \(\displaystyle \frac{1}{3}\).
Answer: \(\displaystyle \frac{16}{3}\).
(d)
Add the two radial boundaries to the arc length.
The two straight radii contribute 8 cm.
Answer: \(8\).
Example 2 · Simple sectors and arc fractions
A sector has radius 5 cm and central angle \(90\)°.
Find the fraction of a complete circle occupied by the sector.
Write its arc length as kπ cm. Find k.
Write its area as \(\displaystyle k\pi \mathrm{cm}^{2}\). Find k.
Its perimeter is aπ+b cm. Find b.
Show the reasoning
(a)
Express the central angle as a fraction of a full turn.
Divide \(90\) by 360 to obtain \(\displaystyle \frac{1}{4}\).
Answer: \(\displaystyle \frac{1}{4}\).
(b)
Apply the angle fraction to circumference.
The circle circumference coefficient 10 is multiplied by \(\displaystyle \frac{1}{4}\).
Answer: \(\displaystyle \frac{5}{2}\).
(c)
Apply the angle fraction to circle area.
The circle area coefficient 25 is multiplied by \(\displaystyle \frac{1}{4}\).
Answer: \(\displaystyle \frac{25}{4}\).
(d)
Add the two radial boundaries to the arc length.
The two straight radii contribute 10 cm.
Answer: \(10\).
Example 3 · Simple sectors and arc fractions
A sector has radius 6 cm and central angle \(72\)°.
Find the fraction of a complete circle occupied by the sector.
Write its arc length as kπ cm. Find k.
Write its area as \(\displaystyle k\pi \mathrm{cm}^{2}\). Find k.
Its perimeter is aπ+b cm. Find b.
Show the reasoning
(a)
Express the central angle as a fraction of a full turn.
Divide \(72\) by 360 to obtain \(\displaystyle \frac{1}{5}\).
Answer: \(\displaystyle \frac{1}{5}\).
(b)
Apply the angle fraction to circumference.
The circle circumference coefficient 12 is multiplied by \(\displaystyle \frac{1}{5}\).
Answer: \(\displaystyle \frac{12}{5}\).
(c)
Apply the angle fraction to circle area.
The circle area coefficient 36 is multiplied by \(\displaystyle \frac{1}{5}\).
Answer: \(\displaystyle \frac{36}{5}\).
(d)
Add the two radial boundaries to the arc length.
The two straight radii contribute 12 cm.
Answer: \(12\).
Common mistakes
Keep intermediate values exact until the final step.