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.

angle_definition.jpg

 

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.

zero_angle.jpg

 

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.

one_right_angle.jpg

 

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.

straight_angle.jpg

 

  • 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:

coterminal_angles.jpg

 

-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:

radian_definition.jpg

 

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).

arc_and_sector.jpg

 

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.
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