Arcs, Sectors and Segments: Quick Revision

A brief note on Arcs, Sectors and Segments. Read it before a test. Try each question at the end before you open its answer.

Arcs and sectors

• Arc length: $l = \dfrac{\theta}{360} \times 2\pi r$, with $\theta$ in degrees. In radians, $l = r\theta$.
• Perimeter of a sector: $l + 2r$.
• Area of a sector: $A = \dfrac{\theta}{360} \times \pi r^2 = \dfrac{1}{2}\,l\,r$.
• The minor and the major sector add up to the whole circle. The major sector has the angle $360^\circ - \theta$.
• A wiper blade between the radii $r$ and $R$ sweeps $\dfrac{\theta}{360}\,\pi\,(R^2 - r^2)$.

Segments

• Minor segment: the sector minus the triangle $OAB$: $\dfrac{\theta}{360}\pi r^2 - \dfrac{1}{2}r^2\sin\theta$.
• Major segment: the whole circle minus the minor segment.
• For $\theta = 90^\circ$ the triangle is $\dfrac{1}{2}r^2$. For $\theta = 60^\circ$ it is $\dfrac{\sqrt{3}}{4}r^2$.
• Use $\pi = \dfrac{22}{7}$ only when told, and show the exact form with $\pi$.

A worked example

A chord subtends $60^\circ$ at the centre of a circle of radius $10\ \text{cm}$. Find the area of the minor segment.

1. The sector is $\dfrac{1}{6}\pi \times 100 = \dfrac{50\pi}{3}\ \text{cm}^2$.
2. The triangle is equilateral: $\dfrac{\sqrt{3}}{4} \times 100 = 25\sqrt{3}\ \text{cm}^2$.
3. The segment is $\left(\dfrac{50\pi}{3} - 25\sqrt{3}\right)\ \text{cm}^2$, which is $9.06\ \text{cm}^2$.

Practice with new numbers

A program to try

The program finds the arc, the sector and both segments for the worked example.

Mistakes to avoid

• Giving the sector as the segment.
• Using $2\pi r$ for an area. The area has $r^2$.
• Forgetting the two radii in the perimeter of a sector.
• Using degrees in $l = r\theta$. That rule needs radians.

Check yourself

1. An arc makes $90^\circ$ at the centre of a circle of radius $14\ \text{cm}$. Take $\pi = \dfrac{22}{7}$. How long is it?

2. A sector of a circle of radius $6\ \text{cm}$ has an angle of $60^\circ$. Write its area in terms of $\pi$.

3. A sector has an arc of $20\ \text{cm}$ and a radius of $5\ \text{cm}$. What is its area?

4. The minor segment of a circle has an area of $12\ \text{cm}^2$ and the circle has an area of $50\ \text{cm}^2$. What is the major segment?