Arc Length and Area of a Sector
A pizza in a restaurant in Galway is cut into 8 equal slices. How much crust does each slice have, and how much cheese? Each slice is a fraction of the whole circle, so you only need to find that fraction and apply it to the circle's circumference and area. By the end of this page you will have answered both.
Sectors and the fraction of a circle
A sector is the region of a circle between two radii and the arc that joins their ends. A slice of pizza, a wedge of cake or the area swept by a windscreen wiper are all sectors.
The angle between the two radii is the central angle $\theta$. A full turn is $360^\circ$, so a sector takes up the fraction
$$\frac{\theta}{360}$$of the whole circle.
A circle has radius $6$ cm. A sector has a central angle of $60^\circ$. Find the length of the arc, in terms of $\pi$.
A circle has radius $6$ cm. A sector has a central angle of $60^\circ$.
Find the length of the arc, in terms of $\pi$.
Arc length and area
The circumference of a circle is $2\pi r$ and its area is $\pi r^2$. Take the fraction $\frac{\theta}{360}$ of each:
The perimeter of a sector is not just the arc. It includes the two straight radii:
$$P = l + 2r$$Leave answers "in terms of $\pi$" when asked: keep the $\pi$ as a symbol and simplify the fraction.
A circle has radius $10$ cm. A sector has a central angle of $72^\circ$. Find the area of the sector, in terms of $\pi$.
A circle has radius $10$ cm. A sector has a central angle of $72^\circ$.
Find the area of the sector, in terms of $\pi$.
A circular pizza in a restaurant in Galway has a radius of $15$ cm. It is cut into $8$ equal slices. a) Find the central angle of each slice. b) Find the area of one…
A circular pizza in a restaurant in Galway has a radius of $15$ cm. It is cut into $8$ equal slices.
- a) Find the central angle of each slice.
- b) Find the area of one slice, correct to one decimal place.
- c) Find the perimeter of one slice, correct to one decimal place.
Working backwards
Sometimes you know the arc length, the area or the perimeter, and you need the angle or the radius. Write the formula, substitute what you know, and solve the equation for the unknown.
A sector of a circle of radius $5$ cm has an arc length of $7$ cm. Find the central angle of the sector, correct to the nearest degree.
A sector of a circle of radius $5$ cm has an arc length of $7$ cm.
Find the central angle of the sector, correct to the nearest degree.
A sector of a circle has a radius of $12$ cm and an area of $48\pi$ cm$^2$. a) Find the central angle of the sector. b) Find the length of the arc, in terms of $\pi$. c)…
A sector of a circle has a radius of $12$ cm and an area of $48\pi$ cm$^2$.
- a) Find the central angle of the sector.
- b) Find the length of the arc, in terms of $\pi$.
- c) Find the perimeter of the sector, correct to one decimal place.
A sector of a circle has a perimeter of $30$ cm and a radius of $8$ cm. a) Find the arc length. b) Find the central angle, correct to the nearest degree. c) Find the…
A sector of a circle has a perimeter of $30$ cm and a radius of $8$ cm.
- a) Find the arc length.
- b) Find the central angle, correct to the nearest degree.
- c) Find the area of the sector, correct to one decimal place.
Combined shapes: wiper regions and segments
Many problems join a sector with another shape.
Region swept by a wiper. If a blade covers distances from $10$ cm to $50$ cm from the pivot, the swept region is a big sector minus a small one. Find both areas with the same angle and subtract.
Segment. The region between a chord and its arc is a segment. Cut it out as: sector $-$ triangle.
The area of a triangle with two sides $a$, $b$ and the angle $C$ between them is $\frac12 ab \sin C$. For triangle $AOB$ with two sides equal to the radius, that is $\frac12 r^2 \sin\theta$.
A windscreen wiper has a blade of length $40$ cm attached to a pivot. The wiper sweeps through an angle of $110^\circ$. The blade extends from $10$ cm to $50$ cm from…
A windscreen wiper has a blade of length $40$ cm attached to a pivot. The wiper sweeps through an angle of $110^\circ$. The blade extends from $10$ cm to $50$ cm from the pivot.
- a) Find the area of the region swept by the blade, correct to the nearest cm$^2$.
- b) Find the distance travelled by the tip of the blade (the point $50$ cm from the pivot) in one sweep, correct to the nearest cm.
A circle with centre $O$ has radius $10$ cm. Points $A$ and $B$ lie on the circle and $\angle AOB = 80^\circ$. a) Find the area of the sector $AOB$, correct to one…
A circle with centre $O$ has radius $10$ cm. Points $A$ and $B$ lie on the circle and $\angle AOB = 80^\circ$.
- a) Find the area of the sector $AOB$, correct to one decimal place.
- b) Find the area of triangle $AOB$, correct to one decimal place.
- c) Hence find the area of the segment between the chord $AB$ and the arc $AB$.
A sector of radius $r$ cm has central angle $\theta^\circ$. The arc length is equal to the radius. a) Show that $\theta = \dfrac{180}{\pi}$. b) Hence show that the area…
A sector of radius $r$ cm has central angle $\theta^\circ$. The arc length is equal to the radius.
- a) Show that $\theta = \dfrac{180}{\pi}$.
- b) Hence show that the area of the sector is $\dfrac{r^2}{2}$.
- c) Find $r$ if the area of the sector is $50$ cm$^2$.
Back to the pizza: each slice has about $11.8$ cm of crust and an area of $88.4$ cm².
Summary
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A sector with central angle $\theta$ is $\frac{\theta}{360}$ of a circle.
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Arc length: $l = \frac{\theta}{360} \times 2\pi r$. Area: $A = \frac{\theta}{360} \times \pi r^2$.
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Perimeter of a sector: $l + 2r$. Check: $A = \frac12 lr$.
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To work backwards, substitute what you know and solve for $\theta$ or $r$.
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Swept region: subtract sectors. Segment: sector minus triangle ($\frac12 r^2 \sin\theta$).