Area of Sector

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Area of Sector

Recall, the area of a circle is given as \( \scriptsize \pi r^2 \)

In general, the area of a sector of a circle is proportional to the angle of the sector as shown in the diagram above, i.e. the area of the sector XOY is \( \frac{\theta}{360^o} \)of the whole circle, that is:

Area of Sector XOY = \( \frac {\theta}{360} \scriptsize \: \times \: \pi r^2 \)

Example 8.2.1:

A pie chart is divided into four sectors as shown in the diagram below. Each sector represents a percentage of the whole. The two larger sectors are equal, and each represents X%. What is the angle subtended by one of those larger sectors? (WAEC)

Solution:

Note: x% + x% + 21 + 9 = 100%

i.e. 2x% = 100 – 30

i.e. x% = \( \frac{70}{2}\)= 35%

x% = 35%

by proportions:

35% ≡ x°
100% ≡ 360º

x° = \( \scriptsize 360^o \: \times \: \normalsize \frac{35}{100} \\ = \scriptsize 36\not{0}^{o} \: \times \: \normalsize \frac{35}{10\not{0}} \\ = \scriptsize 18^o \: \times \: \normalsize \frac{35}{5} \\ = \scriptsize 18^o \: \times \: 7^o \\ = \scriptsize 126^o \)

Example 8.2.2:

In the diagram below, ABCD is a rhombus with dimensions as shown. BXD is a circular arc with centre A. Calculate the area of the blue shaded section to the nearest cm^2.