Arc length and sector area

Radius sprayed

\(\displaystyle 5\text{ m}\)

Angle swept

\(\displaystyle 144^\circ\)

A lawn sprinkler sprays out to 5 m while it sweeps through an angle of 144°. What is the area of lawn watered?

The shaded sector is the part of the circle being measured. Its solid curved edge is the arc; the dashed edge completes the whole circle. Change the central angle, then the radius. Compare the arc length and sector area with the whole circle

What happened when the angle was doubled from 60° to 120° with the radius fixed?

Arc length and sector area

In the first example, the radius is 5 m and the central angle is 144°. In the major-piece example, the stated minor angle is 60°. A sector is a fraction of its circle. The fraction is the central angle out of 360 degrees, and the arc and the area are that same fraction of the circumference and the circle area

1. 1Find the circle fraction Divide the central angle by 360 degrees to find the fraction of a full turn \(\displaystyle \frac{144}{360}=\frac{2}{5}\)
2. 2Use the arc formula The arc is the same fraction of the circumference, which uses one radius \(\displaystyle s=\frac{2}{5}\times2\pi(5)=4\pi\text{ m}\)
3. 3Use the sector-area formula The sector is the same fraction of the whole circle area, which uses the radius squared \(\displaystyle A=\frac{2}{5}\times\pi(5)^2=10\pi\text{ m}^2\)
4. 4Find the major angle For a major piece, subtract the stated minor angle from 360 degrees before applying a formula \(\displaystyle \begin{gathered}\theta_{\mathrm{major}}=360^\circ-60^\circ\\\theta_{\mathrm{major}}=300^\circ\end{gathered}\)

Diameter of the bed

\(\displaystyle 12\text{ m}\)

Paved angle

\(\displaystyle 60^\circ\)

A circular garden bed has a diameter of 12 m. A 60° minor sector is paved and the shaded remainder is lawn. Find the curved edge of the lawn, in metres to 2 decimal places

The lawn's angle is what is left of the full turn

Diameter of the bed

\(\displaystyle 12\text{ m}\)

Paved angle

\(\displaystyle 60^\circ\)

Find the area of the shaded lawn, in square metres to 2 decimal places

Use the same lawn angle as before, with the radius from half the diameter

Choose the required piece before calculating

A minor angle labels the smaller piece. If the question asks for the major piece, first subtract that angle from 360°

1. 1Minor piece Use its stated angle as a fraction of 360° \(\displaystyle \frac{60}{360}=\frac{1}{6}\)
2. 2Major piece The remaining angle is 300°. Use five sixths of the full circumference or area \(\displaystyle \begin{gathered}360^\circ-60^\circ=300^\circ\\\frac{300}{360}=\frac{5}{6}\end{gathered}\)
3. 3Exact or rounded An exact answer keeps π. A decimal answer is rounded once at the end

A student finds the area of a 45° sector with radius 4 cm. Which is the first incorrect line?