Arc length and sector area practice questions | General Mathematics (Year 11, VCE Units 1 & 2)

Arc length and sector area

For a circle with radius 6 cm and central angle 120°, the examples below show the arc length and sector area. A central angle takes a fraction of one full circle. Use the radius once for arc length and the squared radius for sector area

\begin{gathered}s=\frac{\theta}{360}\times2\pi r\\A=\frac{\theta}{360}\times\pi r^{2}\end{gathered}

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 arc formula Arc length is the same fraction of the circumference and contains one radius \(\displaystyle s=\frac{120}{360}\times2\pi(6)=4\pi\text{ cm}\)
Use the sector-area formula Sector area is the same fraction of the circle area and uses the radius squared \(\displaystyle \begin{gathered}A=\frac{120}{360}\times\pi(6)^2\\A=12\pi\text{ cm}^2\end{gathered}\)
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 this fraction 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 is only the central angle's fraction of the full circle, so the full-circle area is too large

Swapping 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 the circumference formula uses one radius and gives a length

Using 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 angle

Difficulty levels

introFind a minor arc length when the radius is given

standardFind a minor sector area or both measures using a stated radius

challengeFind a major arc or sector measure when a diameter is given, sometimes in exact π form

examChoose the major-piece measure from values containing fraction, formula and diameter errors

Questions about this topic

Why is the angle divided by 360?+

A full turn around a circle is 360 degrees, so a central angle covers the same fraction of the circle as it does of a full turn

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 11, VCE Units 1 & 2, General Mathematics, chapter 10: Measurement and similarity

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Arc length and sector area