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Revision:Radians

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TSR Wiki > Study Help > Subjects and Revision > Revision Notes > Mathematics > Radians

1 Introduction
2 Arc Length
3 Area of Sector
4 Comments

Introduction

Radians, like degrees, are a way of measuring angles.

One radian is equal to the angle formed when the arc opposite the angle is equal to the radius of the circle. So in the above diagram, the angle ø is equal to one radian since the arc AB is the same length as the radius of the circle.

Due to their usefulness, people often omit the symbol for radians. It is written in a similar way to degrees, ie one radian would be written $1^{c}$ .

Now, the circumference of the circle is $2\pi r$ , where r is the radius of the circle. So the circumference of a circle is $2\pi$ larger than its radius. This means that in any circle, there are $2\pi^{c}$ Therefore $360^{\circ} = 2\pi^{c}$ Therefore $180^{\circ} = \pi^{c}$ So $1^{c} = \frac{180}{\pi}^{\circ}$ and $1^{\circ} = \frac{\pi}{180}^{c}$

Therefore to convert a certain number of degrees in to radians, multiply the number of degrees by p/180 (for example, 90º = 90 × p/180 radians = p/2). To convert a certain number of radians into degrees, multiply the number of radians by 180/p .

Arc Length

The length of an arc of a circle is equal to rø, where ø is the angle, in radians, subtended by the arc at the centre of the circle. So in the below diagram, s = rø .

(Diagram missing)

Area of Sector

The area of a sector of a circle is ½ r² ø, where r is the radius and ø the angle in radians subtended by the arc at the centre of the circle. So in the below diagram, the shaded area is equal to ½ r² ø .

(Diagram missing)