You should know with 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 \( (1,-2) \).
Radius of a Circle
A radius of a circle 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 an arc smaller than a semicircle and is also defined as the shorter arc connecting two endpoints of a diameter. A central angle that is subtended by a minor arc has a measure less than \( 180^{\circ} \).
Major Arc of a Circle
A major arc is an arc larger than a semicircle and is also defined as the larger arc connecting two endpoints of a diameter. A central angle that is 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 circumference of a circle is \( C = 2 \pi r \), as the centre angle is \( 2 \pi \).
Correspondingly, when the center angle is \( \theta \), the arc, which is a part of the circumference, is calculated as;
$$ \ell = 2 \pi \times \displaystyle \frac{\theta}{2 \pi r} = \theta r $$
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 $30^{\circ}$.
\( \begin{align} \displaystyle
30^{\circ} &= 30^{\circ} \times \dfrac{\pi}{180^{\circ}} \\
&= \dfrac{\pi}{6} \text{ radians}\\
\text{arc length } \ell &= 4 \times \dfrac{\pi}{6} \\
&= \dfrac{3 \pi}{2} \text{ cm}\\
\text{sector area }A &= \dfrac{1}{2} \times \dfrac{\pi}{6} \times 4^2 \\
&= \dfrac{2 \pi}{3} \text{ cm}^2
\end{align} \)
Realted Topics of Arc Length and Sector Area
Algebra Algebraic Fractions Arc Binomial Expansion Capacity Common Difference Common Ratio Differentiation Divisibility Proof 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 Proof Pythagoras Theorem Quadratic Quadratic Factorise Rational Functions Sequence Sketching Graphs Surds Time Transformation Trigonometric Functions Trigonometric Properties Volume
