Angle and Its Measurement

• An Angle

An angle is a figure obtained, when 2 rays share a common starting point. The point is known as the vertex. The rays are termed as initial arm and final arm of the angle.

In the figure above, point $O$ is the vertex, ray $OA$ is the initial arm, ray $OB$ is the final arm.

• Measuring the Angle

The measure of an angle is the amount of rotation of the terminal arm from the initial arm. If the rotation is clockwise, the measure is taken to be negative and if the rotation is anti-clockwise, the measure is taken to be positive.

• Directed Angle

A directed angle distinguishes between 2 angles of equal magnitude on the basis of sense of rotation. So, in the figure above, $\angle AOB$ is different from $\angle BOA$. For $\angle AOB$, the initial arm is $OA$ and final arm is $OB$. For $\angle BOA$, $OB$ is the initial arm and $OA$ is the final arm.

• Zero, Right and Straight Angles

A zero angle is the one, whose initial and terminal arms coincide and rays point in the same direction.

A right angle is formed, when the initial and terminal arms are perpendicular to each other. This angle is easily recognizable to our eyes. $\angle PQR$ is a right angle, where $QP$ and $QR$ are perpendicular to each other.

A straight angle is the one, whose initial and terminal arms form a straight line. The magnitude of straight angle is double that of right angle. In other words, it is made up of 2 right angles. In the figure below, arms $OA$ and $OB$ form a straight line; hence $\angle BOA$ is a straight angle.

• Angle Measurement in Degrees

To measure the amount of rotation of terminal ray, we have sexagesimal system, where 1 right angle is equal to $90^o$. Thus, a straight angle measures $180^o$ and a zero angle measures $0^o$.

The angle of $1^o$ is the $\frac {1}{90}th$ part of a right angle. $1^o$ is further divided into $60$ equal parts. $\frac {1}{60}th$ part of $1^o$ is known as $1 \ minute$ and is denoted by $1^{'}$. The angle of $1^{'}$ is further divided into $60$ equal parts and each part has magnitude $1 \ second$ and is denoted by $1^{''}$. Thus,

${1^{'}= \frac {1}{60}^{o} \ and \ 1^{''} = \frac {1}{60}^{'}}$

When the terminal ray is rotated anticlockwise through 1 complete revolution, the measure of the new angle is $360^o$. Note that this angle and the zero angle share same initial and terminal arms.

• Coterminal Angles

The coterminal angles are the ones, which share same initial and terminal arms. These are shown below:

$-315^o, 45^o$ and $405^o$ are all coterminal angles. If an angle has a measure of $M^o$, all angles with measure $M^o \pm i \times 360^o$ are its coterminal angles. $i$ is any integer.

• Angle Measurement in Radians

There is another system of unit to measure an angle, which known as circular system. The unit angle in this system is $1 \ radian$ and is denoted by $1^c$. It is defined as follows:

Consider a circle with center $C$ and radius $r$. Recall that the circumference of the circle is equal to $2 \pi r$. Now consider an arc shown by dotted lines, whose length is equal to the radius of the circle. The angle subtended by this arc at the center of the circle is $1^c$.

The definition of $1^c$ is independent of the radius of circle.

• Conversion between Degrees and Radians

The inter-conversion between degrees and radians is governed by the following relation:

${180^o = \pi^c, \ 1^o = {\frac {\pi}{180}}^c, \ 1^c = \frac {180}{\pi}}$

$1^c$ is approximately equal to $57^o$.

• Length of an Arc and Area of a Sector

Consider a circle with radius $r$ and an arc of this circle, say $AB$. The angle subtended by $AB$ at $C$ is $\theta$ in radians. The region bounded by the arc $AB$ and the radii $CA$ and $CB$ is known as a sector of the circle (the shaded region in the figure).

The length $l$ of arc is given by

${l = r \theta}$

and the area of the sector is given by

${A = \frac 1 2 r^2 \theta}$

• Useful Points

1. The sum of all angles of a triangle is $180^o$ or $\pi^c$.
2. The sum of all angles of a quadrilateral is $360^o$ or $2 \pi^c$.
3. A regular polygon is the one, whose sides are equal in length.
4. Perimeter of any closed figure is the sum of lengths of its sides.