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Solved Examples on the Areas of Sectors

Samuel Dominic Chukwuemeka (SamDom For Peace) Verify your answers with these Calculators as applicable.

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$ d = diameter \\[3ex] r = radius \\[3ex] A = area\;\;of\;\;sector \\[3ex] L = length\;\;of\;\;arc \\[3ex] \theta = central\:\:angle \\[3ex] \pi = \dfrac{22}{7} \\[5ex] RAD = radians \\[3ex] ^\circ = DEG = degrees \\[3ex] A = \dfrac{\pi r^2\theta}{360} ...\theta \:\:in\:\:DEG \\[5ex] \theta = \dfrac{360A}{\pi r^2} ...\theta \:\:in\:\:DEG \\[5ex] r = \dfrac{360A}{\pi\theta} ...\theta \:\:in\:\:DEG \\[5ex] A = \dfrac{r^2\theta}{2} ...\theta \:\:in\:\:RAD \\[5ex] \theta = \dfrac{2A}{r^2} ...\theta \:\:in\:\:RAD \\[5ex] r = \sqrt{\dfrac{2A}{\theta}} ...\theta \:\:in\:\:RAD \\[5ex] Circumference\:\:of\:\:a\:\:circle = 2\pi r = \pi d \\[3ex] A = \dfrac{Lr}{2} \\[5ex] r = \dfrac{2A}{L} \\[5ex] L = \dfrac{2A}{r} $

(1.)


This implies that:

$ r = 3\dfrac{1}{5} = \dfrac{16}{5} \\[5ex] L = 4 \\[3ex] \theta = \dfrac{L}{r} \\[5ex] \theta = 4 \div \dfrac{16}{5} \\[5ex] \theta = 4 * \dfrac{5}{16} \\[5ex] \theta = \dfrac{4 * 5}{16} \\[5ex] \theta = \dfrac{20}{16} = \dfrac{5}{4} RAD $
(2.) JAMB A sector of a circle has an area of $55\;cm^2$
If the radius of the circle is $10\;cm$, calculate the angle of the sector.

$ \left[\pi = \dfrac{22}{7}\right] \\[5ex] A.\;\; 45^\circ \\[3ex] B.\;\; 63^\circ \\[3ex] C.\;\; 75^\circ \\[3ex] D.\;\; 90^\circ \\[3ex] $

$ A = \dfrac{\pi r^2\theta}{360} \\[5ex] 360A = \pi r^2 \theta \\[3ex] \pi r^2 \theta = 360A \\[3ex] \theta = 360A \div \pi r^2 \\[3ex] \theta = (360 * 55) \div \left(\dfrac{22}{7} * 10 * 10\right) \\[5ex] \theta = 360 * 55 * \dfrac{7}{22} * \dfrac{1}{10} * \dfrac{1}{10} \\[5ex] \theta = \dfrac{360 * 55 * 7}{22 * 10 * 10} \\[5ex] \theta = \dfrac{36 * 5 * 7}{2 * 1 * 10} \\[5ex] \theta = \dfrac{18 * 7}{2} \\[5ex] \theta = 9 * 7 \\[3ex] \theta = 63^\circ $
(3.)


$ L = \dfrac{2\pi r\theta}{360} ...\theta \:\:in\:\:DEG \\[5ex] \pi = \dfrac{22}{7} \\[5ex] r = 6400 km \\[3ex] \theta = 12^\circ \\[3ex] L = \left(2 * \dfrac{22}{7} * 6400 * 12\right) \div 360 \\[5ex] L = 2 * \dfrac{22}{7} * 6400 * 12 * \dfrac{1}{360} \\[5ex] L = \dfrac{2 * 22 * 6400 * 12}{7 * 360} \\[5ex] L = \dfrac{3379200}{2520} \\[5ex] L = 1340.952381 \\[3ex] L \approx 1341 \:km $
(4.) JAMB An arc subtends an angle $50^\circ$ at the centre of circle of radius $6\;cm$
Calculate the area of the sector formed.

$ \left[\pi = \dfrac{22}{7}\right] \\[5ex] A.\;\; \dfrac{100}{7}\;\;cm^2 \\[5ex] B.\;\; \dfrac{110}{7}\;\; cm^2 \\[5ex] C.\;\; \dfrac{80}{7}\;\;cm^2 \\[5ex] D.\;\; \dfrac{90}{7}\;\;cm^2 \\[5ex] $

$ A = \dfrac{\pi r^2\theta}{360} ...\theta \:\:in\:\:DEG \\[5ex] A = (\pi * r^2 * \theta) \div 360 \\[5ex] A = \dfrac{22}{7} * \dfrac{6}{1} * \dfrac{6}{1} * \dfrac{50}{1} * \dfrac{1}{360} \\[5ex] A = \dfrac{22 * 5}{7} \\[5ex] A = \dfrac{110}{7}\;\; cm^2 $
(5.)


(6.) JAMB A circular arc subtends angle $150^\circ$ at the centre of a circle of radius $12\;cm$
Calculate the area of the sector of the arc.

$ A.\;\; 30\pi\;cm^2 \\[3ex] B.\;\; 60\pi\;cm^2 \\[3ex] C.\;\; 120\pi\;cm^2 \\[3ex] D.\;\; 150\pi\;cm^2 \\[3ex] $

$ A = \dfrac{\pi r^2\theta}{360} ...\theta \:\:in\:\:DEG \\[5ex] A = (\pi * r^2 * \theta) \div 360 \\[5ex] A = \dfrac{\pi}{1} * \dfrac{12}{1} * \dfrac{12}{1} * \dfrac{150}{1} * \dfrac{1}{360} \\[5ex] A = \dfrac{\pi * 12 * 15}{3} \\[5ex] A = \pi * 12 * 5 \\[3ex] A = 60\pi\;\; cm^2 $
(7.)


(8.) JAMB A chord of a circle subtends an angle of $120^\circ$ at the centre of a circle of diameter $4\sqrt{3}\;cm$
Calculate the area of the major sector.

$ A.\;\; 8\pi\;cm^2 \\[3ex] B.\;\; 32\pi\;cm^2 \\[3ex] C.\;\; 4\pi\;cm^2 \\[3ex] D.\;\; 16\pi\;cm^2 \\[3ex] $

$ 2r = d \\[3ex] r = \dfrac{d}{2} \\[5ex] d = 4\sqrt{3} \\[3ex] r = \dfrac{4\sqrt{3}}{2} = 2\sqrt{3} \\[5ex] $ Let us draw the diagram
Number 8

$ \underline{Major\;\;sector\;\;ADB} \\[3ex] Reflex\;\; \angle \phi + 120 = 360 ...\angle s \;\;at\;\;a\;\;point \\[3ex] \phi = 360 - 120 = 240 \\[3ex] \phi = 240^\circ = \dfrac{240}{180}\pi = \dfrac{4}{3}\pi = \dfrac{4\pi}{3} \\[5ex] A = \dfrac{r^2\phi}{2} ...\phi \:\:in\:\:RAD \\[5ex] A = \dfrac{1}{2} * r^2 * \phi \\[3ex] A = \dfrac{1}{2} * \left(2\sqrt{3}\right)^2 * \dfrac{4\pi}{3} \\[5ex] A = \dfrac{1}{2} * 4 * 3 * \dfrac{4\pi}{3} \\[5ex] A = 2 * 4\pi \\[3ex] A = 8\pi cm^2 $