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 is the vertex, ray is the initial arm, ray 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.
A directed angle distinguishes between 2 angles of equal magnitude on the basis of sense of rotation. So, in the figure above, is different from . For , the initial arm is and final arm is . For , $OB$ is the initial arm and 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. is a right angle, where and 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 and form a straight line; hence 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 . Thus, a straight angle measures and a zero angle measures .
The angle of is the part of a right angle. is further divided into equal parts. part of is known as and is denoted by . The angle of is further divided into equal parts and each part has magnitude and is denoted by . Thus,
When the terminal ray is rotated anticlockwise through 1 complete revolution, the measure of the new angle is . Note that this angle and the zero angle share same initial and terminal arms.
The coterminal angles are the ones, which share same initial and terminal arms. These are shown below:
and $405^o$ are all coterminal angles. If an angle has a measure of , all angles with measure are its coterminal angles. 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 and is denoted by . It is defined as follows:
Consider a circle with center and radius . Recall that the circumference of the circle is equal to . 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 .
The definition of 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:
is approximately equal to .
Length of an Arc and Area of a Sector
Consider a circle with radius and an arc of this circle, say . The angle subtended by at is in radians. The region bounded by the arc and the radii and is known as a sector of the circle (the shaded region in the figure).
The length $l$ of arc is given by
and the area of the sector is given by
- The sum of all angles of a triangle is or .
- The sum of all angles of a quadrilateral is or .
- A regular polygon is the one, whose sides are equal in length.
- Perimeter of any closed figure is the sum of lengths of its sides.