Arc Length and Sector Area

Arc Length and Sector Area

You should know these terms relating to the parts of a circle.

Centre of a Circle

The centre of a circle is the point that is equidistant from all points on the circle.

centre of a circle

The centre of a circle \( x^2 + y^2 = 9 \) is \( (0,0) \).
The centre of a circle \( (x-1)^2 + y^2 = 9 \) is \( (1,0) \).
The centre of a circle \( x^2 + (y+2)^2 = 9 \) is \( (0,-2) \).
The centre of a circle \( (x-1)^2 + (y+2)^2 = 9 \) is \( (1,-2) \).

The Radius of a Circle

A circle’s radius is a straight line joining the centre of a circle to any point on the circumference.

radius of a circle

The radius of a circle \( x^2 + y^2 = 9 \) is \( r=3 \sqrt{9} = 3 \).
The radius of a circle \( (x-1)^2 + y^2 = 4 \) is \( r = \sqrt{4}=2 \).
The radius of a circle \( x^2 + (y+2)^2 = 16 \) is \( r = \sqrt{16}=4 \).
The radius of a circle \( (x-1)^2 + (y+2)^2 = 100 \) is \( r = \sqrt{100} = 10 \).

Minor Arc of a Circle

A minor arc is smaller than a semicircle and is also defined as a shorter arc connecting two endpoints of a diameter. A central angle subtended by a minor arc has a measure less than \( 180^{\circ} \).

minor arc of a circle

Major Arc of a Circle

A major arc is larger than a semicircle and is also defined as the larger arc connecting two endpoints of a diameter. A central angle subtended by a major arc has a measure larger than \( 180^{\circ} \).

major arc of a circle

Arc Length Formula

The arc length formula is used to find the length of an arc of a circle; $ \ell =\theta r$, where $\theta$ is in radian.

arc length of a circle

The circle’s circumference is \( C = 2 \pi r \), as the centre angle is \( 2 \pi \).
Correspondingly, when the centre angle is \( \theta \), the arc, which is a part of the circumference, is calculated as;

$$ \large \ell = 2 \pi \times \displaystyle \frac{\theta}{2 \pi r} = \theta r $$

Sector Area Formula

Sector area is found $\displaystyle A=\dfrac{1}{2}\theta r^2$, as everyone knows this, where $\theta$ is in radian.

sector area of a circle

Example 1

Find the arc length and area of a sector of a circle of radius $6$ cm and the centre angle $\dfrac{2 \pi}{5}$.

\( \begin{align} \displaystyle
\text{arc length } \ell &= 6 \times \dfrac{2 \pi}{5} \\
&= \dfrac{12 \pi}{5} \text{ cm}\\
\text{sector area } A &= \dfrac{1}{2} \times \dfrac{2 \pi}{5} \times 6^2 \\
&= \dfrac{36 \pi}{5} \text{ cm}^2
\end{align} \)

Example 2

Find the arc length and area of a sector of a circle of radius $4$ cm and the centre angle $30^{\circ}$.

\( \begin{align} \displaystyle
30^{\circ} &= 30^{\circ} \times \dfrac{\pi}{180^{\circ}} \\
&= \dfrac{\pi}{6} \text{ radians} \\
\text{arc length } \ell &= 4 \times \dfrac{\pi}{6} \\
&= \dfrac{3 \pi}{2} \text{ cm} \\
\text{sector area }A &= \dfrac{1}{2} \times \dfrac{\pi}{6} \times 4^2 \\
&= \dfrac{2 \pi}{3} \text{ cm}^2
\end{align} \)

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