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In the study of circles, the radian measure is a way of expressing angles that is directly related to the radius of the circle. It provides a simple relationship between the length of an arc and the angle subtended at the centre of the circle by that arc. Similarly, the area of a sector can be found by knowing the angle in radians and the radius of the circle. This approach to measurement simplifies many mathematical calculations and is fundamental in trigonometry and calculus.
Key Formulae:
The length of an arc \( l \) in a circle of radius \( r \) with a central angle \( \theta \) measured in radians is given by:
\[ l = r\theta \]
The area \( A \) of a sector of a circle with radius \( r \) and central angle \( \theta \) in radians is given by:
\[ A = \frac{1}{2}r^2\theta \]
Example:
Consider a circle with a radius of 4 cm. Calculate the length of the arc and the area of the sector formed by a central angle of 1.5 radians.
The length of the arc is:
\[ l = r\theta = 4 \times 1.5 = 6 \text{ cm} \]
The area of the sector is:
\[ A = \frac{1}{2}r^2\theta = \frac{1}{2} \times 4^2 \times 1.5 = 12 \text{ cm}^2 \]
A nice video explanation of what a radian is (from SuperScript)
This video on Radians, Lengths of Arcs and Areas of Sectors is from Revision Village and is aimed at students taking the IB Mathematics Analysis and Approaches SL and HL courses.
This video on Degrees v Radians is from Revision Village and is aimed at students taking the IB Maths AA SL/HL level course
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