Annenberg – Angle Measurement


While degrees are the most commonly used units of angle measure, there are also other units. For example, angles are sometimes measured in radians in order to simplify certain calculations. The radian measure is defined in the International System of Units (SI) as the ratio of arc length to the radius of the circle. For 1 radian, the arc length is equal to radius:

Screen shot 2013-07-29 at 8.42.04 PM


An angle can be defined as the union of two rays with a common endpoint. (A ray begins at a point and extends infinitely in one direction.) The common endpoint is called the vertex (A in the figure below), and the rays are called the sides of the angle.

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Angle PQR is an angle whose sides are opposite rays. This type of angle is called a straight angle.

Angle QPT is an angle whose sides (PQ and PT) are coincident. This type of angle is called a zero angle.


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  • Polygons A, B, C, G, K, M, and O all have two angles that are supplementary.
  • Polygons D, E, F, and L all have two angles that are complementary.
  • All polygons except L have some congruent angles. In polygons A, B, C, H, I, and N, all angles are congruent!
  • If you take any two polygons and line them up so that one side from each meets at a common vertex, you’ve created a pair of adjacent angles.

The sum of the measures of the angles of a triangle is 180 degrees. Notice in this diagram that the diagonal from one vertex of a quadrilateral to the non-adjacent vertex divides the quadrilateral into two triangles:

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The sum of the angle measures of these two triangles is 360 degrees, which is also the sum of the measures of the vertex angles of the quadrilateral.

In a regular polygon, the measure of each central angle is equal to the measure of each exterior angle:

Screen shot 2013-07-29 at 9.04.38 PM

Since the central angles total 360 degrees and the exterior angles total 360 degrees, each of these angles is also equal. (For non-regular polygons, these totals are still equal, but the individual angles are not.)




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