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1. Angles

by M. Bourne

An angle is a measure of the amount of rotation between two line segments. The 2 line segments (or rays) are named the initial side and terminal side, meeting at a vertex, as shown in the diagram.

If the rotation is anti-clockwise, the angle is positive. Clockwise rotation gives a negative angle (by convention).

Examples of positive and negative angles

Angles are commonly measured in degrees or radians. If you can't wait to learn about radians, see section 7. Radians.

Continues below

Acute, Right, Obtuse, Straight and Reflex angles

There are 5 main types of angles: Acute, Right, Obtuse, Straight and Reflex.

a. Acute angles

An acute angle is between 0o and 90o. The three angles above are all acute angles.

Memory tip: The word "acute" comes from the Latin acutus meaning "sharp", or "pointed".

b. Right angle

90o

A right angle, 90o

A right angle is 90o. We see right angles all the time in the corners of a room, a building or a painting.

Memory tip: The term "right angle " comes from the Latin angulus rectus where rectus means "upright".

c. Obtuse angles

115o

Obtuse angle, 115o

An obtuse angle is between 90o and 180o.

Memory tip: The word "obtuse" comes from the Latin obtusus meaning "dull", "blunted" or "not sharp".

d. Straight angles

180o

Straight angle, 180o

A straight angle is 180o.

e. Reflex angles

206o

Reflex angle, 206o

A reflex angle is between 180o and 360o.

Memory tip: The word "reflex" comes from the Latin reflexus meaning "bending back". A "reflex action" is one where your muscle "bends back" involuntarily.

Standard Position of an Angle

An angle is in standard position if the initial side is the positive x-axis and the vertex is at the origin. All the examples given above are in standard position if the vertex is at (0, 0).

We will use `r`, the length of the hypotenuse, and the lengths x and y when defining the trigonometric ratios on the next page, 2. Sine, Cosine, Tangent & Reciprocals.

initial side terminal side θ r x y (x, y) (0, 0)
x-axis
y-axis

An angle in standard position.

Degrees, Minutes and Seconds

The Babylonians (who lived in modern day Iraq from 5000 BC to 500 BC) used a base `60` system of numbers. From them we get the division of time, latitude & longitude and angles in multiples of `60`.

Similar to the way hours, minutes and seconds are divided, the degree is divided into 60 minutes (') and a minute is divided into 60 seconds ("). We can write this form as: DMS or o ' ".

Exercises

Convert the following:

1) 36o23'47" to decimal degrees

Answer

Your calculator may be able to do this conversion for you directly. The question is similar to asking "How many hours in `36` hours, `23` minutes and `47` seconds?"

What is happening is:

36o = 36o (we don't need to do anything to the whole number of degrees)

23' = `23/60` of 1o = 0.38333o

47" = `47/3600` of 1o = 0.0130555o

Adding them up, we get:

36o + 0.38333o + 0.0130555o = 36.396386o

2) 58.39o to DMS

Answer

We need to convert this to degree-minutes-seconds. Once again, your calculator may be able to do this directly. As always, it is good to know what the calculator is doing for you.

58o = 58o (nothing to do here)

0.39 of 1o = 0.39 × 60' = 23.4'. This means `23` minutes and `0.4` of a minute left over. We still have a decimal portion, so we need to find `0.4` of `1` minute.

0.4 of 1' = 0.4 of 60" = 24".

Putting this together, we have 58.39o = 58o23'24".

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More angles pages

Coming up:

Sin, cos and tan of an angle

Angles using radians

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