Measurement Of Angles Examples

An angle is formed when two rays meet at a common endpoint called the vertex. The amount of rotation from one ray to the other is called the measure of the angle. Angle measurement is an important concept in geometry, trigonometry, engineering, architecture, astronomy, navigation, and everyday problem-solving.

Units of Angle Measurement

The most standard measure is the degree. One complete rotation corresponds to 360°.

Each degree is divided into:

1° = 60 minutes (60′)
1′ = 60 seconds (60″)

Another important unit is the radian, which is based on the geometry of a circle.

One radian is an angle centred on a circle with an arc measure corresponding to the radius. A complete revolution equals:

2π radians = 360°

Therefore,

180° = π radians

Conversion Between Degrees and Radians

To convert degrees to radians:

Radians = Degrees × (π / 180)

Example:
90° = 90 × (π / 180) = π/2 radians

To convert radians to degrees:

Degrees = Radians × (180 / π)

Example:
π/3 radians = (π/3) × (180 / π) = 60°

Types of Angles

The angles are classified depending on their magnitude.

Acute angle: less than 90°
Right angle: exactly 90°
Obtuse angle: between 90° and 180°
Straight angle: exactly 180°
Reflex angle: greater than 180° but less than 360°
Complete angle: exactly 360°

[Image of types of angles including acute, right, obtuse, straight, reflex, and complete angle]

Measuring with a Protractor

A protractor is a semicircular or circular instrument used to measure angles in degrees. To measure an angle using a protractor:

Place the centre of the protractor on the vertex of the angle.
Align the baseline of the protractor with one ray of the angle.
Read the degree marking where the second ray intersects the scale.

Most protractors contain inner and outer scales, allowing angles to be measured in both clockwise and counterclockwise directions.

Example: If one ray passes through 40° on the protractor, the angle measures 40°.

Example: If the ray falls between 110° and 120° near 115°, the angle measures approximately 115°.

Constructing Angles

In order to draw a specific angle, a base ray must be drawn, the protractor's center placed at the end point of the ray, the required degree marked, and the second ray drawn through the marked point.

Angle Addition - If two adjacent angles have a common side and a common vertex, then the angles add up. Example: If angle A is 30° and adjacent angle B is 50°, then the total is 80°.

Linear Pair - Two adjacent angles which form a straight line are equal to 180°.
Example: If one angle is 120°, the other will be 60°, as the two angles add up to 180°.

Vertical Opposite Angles - When two lines are intersecting each other, there is equality between opposite angles.
Example: In case the angle measured 45°, the vertical opposite angle also equals 45°.

Complementary and Supplementary Angles

Complementary angles are two angles whose sum is 90°. Example: 35° + 55° = 90°

Supplementary angles are two angles whose sum is 180°. Example: 120° + 60° = 180°

Angles around a point always add up to 360°.

Example: Suppose three angles around a point measure 90°, 120°, and 80°.
Fourth angle = 360° − (90° + 120° + 80°)
Fourth angle = 360° − 290°
Fourth angle = 70°

Angles in Triangles – The total of interior angles of a triangle is always 180°.

Example: If two angles are 50° and 60° respectively, the third would be 70°.

Exterior Angle Theorem - An exterior angle of a triangle consists of the sum of the other two angles interior to the triangle.

Example: Given the values of the angles 40° and 55°, the exterior angle will be 95°.

Angles in Parallel Lines – When a transversal intersects two parallel lines:

Corresponding angles are equal
Alternate interior angles are equal
Co-interior angles are supplementary (sum = 180°)

Example: If one alternate interior angle measures 75°, the other alternate interior angle will also measure 75°.

Radian Measure Examples

Example: Convert 90° to radians: multiply by π divided by 180, giving π/2.
Example: Convert π radians to degrees: multiply by 180 divided by π, giving 180°.

Real-Life Angle Measurement

Most often, angles are involved in problems about turns in a road, the angle of a roof, the position of the hour and minute hands on a clock, and the direction in which you are headed on a journey. Surveyors measure angles on land, engineers compute angles in their work, and pilots and sailors must know bearings, which are measured in angles of degrees.

Example: At 3:00, the minute hand points at 12 and the hour hand points at 3. The angle between them is 90°, which is a right angle.

Example: If the bearing is 120 degrees, turn 120 degrees in a clockwise manner.

Angle of Elevation and Depression - Measures the upward or downward direction. Example 17: If a person looks up at a 30° angle to see the top of a tower, that is elevation.