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.

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.

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 \( 180^{\circ} \).

## 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 \( 180^{\circ} \).

## 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.

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.

### 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} \)

## Related Topics of Arc Length and Sector Area

Algebra Algebraic Fractions Arc Binomial Expansion Capacity Common Difference Common Ratio Differentiation Double-Angle Formula Equation Exponent Exponential Function Factorials Factorise Functions Geometric Sequence Geometric Series Index Laws Inequality Integration Kinematics Length Conversion Logarithm Logarithmic Functions Mass Conversion Mathematical Induction Measurement Perfect Square Perimeter Prime Factorisation Probability Product Rule Proof Quadratic Quadratic Factorise Ratio Rational Functions Sequence Sketching Graphs Surds Time Transformation Trigonometric Functions Trigonometric Properties Volume

Yⲟu need to be a part of a contest for one of thе most

useful blogs on the net. I’m going to highly rеcommend this websitе!

Tһanks for finally wгiting аbߋut > Arc Length and

Sector Area – iitutоr < Loved it!