Measuring angles: The sexagesimal measure.
First two years of secondary education.


An angle is the region formed by two line segments that extend from a common point.

In the following diagram you can see how the two line segments form two regions and therefore form two angles, A and B. Angle A is a convex angle whereas B is a concave angle.

angulo.gif (3068 bytes)

Some special angles:

A null angle is formed by two straight lines that coincide. It doesn't cover any of the plane.

A right angle is a convex angle formed by two perpendicular straight lines.

A straight angle is formed by two rays (half-lines) which are on the same line but going in opposite directions.

A complete revolution forms an angle which contains the whole plane (360º).

A convex angle is always less than 180º. On the other hand, a concave angle is always greater than 180º

An acute angle is less than a right angle (90º).

An obtuse angle is a convex angle (less than 180º) which is greater than a right angle (90º).

Measuring angles: the sexagesimal measure.

A sexagesimal degree, or just a degree (1º) is the measurement of the angle formed by dividing a right angle into 90 equal parts. Therefore, a right angle measures 90º.

1. In this exercise construct angles of 25º, 135º, 45º, 123º, 180º, 90º, 190º, 0º,  270º, 330º, 360º. Write what kind of angle each one is in your exercise book.

You can change the size of the angle by using the arrows or writing the new value for the angle directly into the space and pressing the Enter key.

The Init button will restore the initial values.

 The protractor

The protractor is a very useful instrument which we can use to construct and measure angles.transporta.gif (4520 bytes)

It is a semicircle, which is marked in degrees around the circumference, and allows us to measure convex angles (up to 180º)

Drag the red point in the following window and look carefully at the angle measurement.

You can change the angle by dragging the the red point with the mouse.

The Init button will restore the initial values.

2. Use your protractor to construct the angles given in the exercise above. Find out how we can use a protractor to construct concave angles.

Parts of a degree.

There are two ways to measure an angle with greater accuracy: Firstly, using the decimal system. This consists of measuring the decimal parts of a degree, which is what a protractor does. The second way is using the sexagesimal measure. This consists of dividing the degree into 60 equal parts, called minutes. Therefore, each degree is 60 minutes (60') and each minute is 60 seconds (60'').

3. Construct these angles in this window and write down if they are acute or obtuse angles into your exercise book:

    a. 56º 20' 40"

    b. 125º 15' 30"

    c. 18º 0' 0''

    d. 18º 35' 10''

    e. 18º 36' 0"

You can change the size of the angle by using the arrows or writing the new value for the angle directly into the space and pressing the ENTER key.

You should notice that there is very little difference between the angles in c, d and e in the exercise above. However, you will be looking at situations when it is important to measure angles very accurately in a later lesson. 

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  Fernando Arias Fernández-Pérez
Spanish Ministry of Education. Year 2001

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