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.
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 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 Pythagoras Theorem Quadratic Quadratic Factorise Ratio Rational Functions Sequence Sketching Graphs Surds Time Transformation Trigonometric Functions Trigonometric Properties Volume
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