Measuring Angles - Practice Questions & MCQ

Measurement of Angle

To form an angle, we start with two rays lying on top of one another. We leave one fixed in place and rotate the other. The fixed ray is the initial side, and the rotated ray is the terminal side. And, the measure of an angle is the amount of rotation from the initial side to the terminal side.

An angle is in standard position if its vertex is located at the origin, and its initial side extends along the positive x-axis, as you can see from the figure given below.

If the angle is measured in a counterclockwise direction from the initial side to the terminal side, the angle is said to be a positive angle. If the angle is measured in a clockwise direction, the angle is said to be a negative angle. 

There are three systems used for the measurement of angles

1. Sexagesimal system

In this system, an angle is measured in degrees, minutes and seconds.
1 Right angle $={90}^{\circ }$ (Read as 90 degrees)
$1^{\circ }={60}^{\prime }(1$ degree $=60$ minutes $)$
$1^{\prime }={60}^{\prime \prime }$ ( 1 minutes $=60$ seconds $)$
2. Centesimal system

In this system, the measurement of the right angle is divided into 100 equal parts, and parts called Grades.
1 Right angle $=100\mathrm{g}($ Read as 100 grades $)$
3. Circular system

In this system, an angle is measured in radians.
One radian is the measure of the central angle of a circle that intercepts an arc equal in length to the radius of that circle. A central angle is an angle formed at the centre of a circle by two radii.

The formula for the radian measure of an angle formed by an arc of length I at the centre of the circle of radius $r$ is (Length of $\mathrm{arc})/($ Radius $)=1/r$

Because the total circumference equals $2\pi$ times the radius, a full circular rotation is $2\pi$ radians.
So, $2\pi$ radians $={360}^{\circ }$
So, $\pi$ radians $={360}^{\circ }/2={180}^{\circ }$
and 1 radian $={180}^{\circ }/\pi \approx {57.3}^{\circ }$

Interconversion of units

Since, degrees and radians both measure angles, we need to be able to convert between them. We can easily do so using a proportion (where θ is the measure of the angle in degrees and θ_R is the measure of the angle in radians)

$\frac{\theta }{180}=\frac{{\theta }_{\mathrm{R}}}{\pi }$
OR

$\frac{\text{ Degree }}{180}=\frac{\text{ Radians }}{\pi }$

Note:

Radian is the unit to measure angles, and it does not mean that $\pi$ stands for ${180}^{\circ }.\pi$ is a real number. Remember the relation, $\pi$ radians $={180}^{\circ }$.

In a circle of radius $r$, the length of an arc is subtended by an angle with measure $\theta$ in radians. Arc length $=($ radius $)x$ (Angle subtended by an arc in radians)

$s=r\theta \text{. }$