Review of Angles
Before we go into angular measurement, a few basic
principles about angles must be covered. An angle can be
described as two lines and a vertex (Figure 1).
The relationship of the two lines can be measured
because the lines intersect. The intersection point
is known as the vertex. Three letters designate angles. ACB in the example represents
Line A, Line B, and Vertex
C. The primary unit of angular measurement is the
degree. There are 360 degrees in a complete circle. Each
degree is divided into 60 parts and these parts are
known as minutes.

Figure 1. The
angle is defined as ACB. It refers to the angle of the
sides, not the space between them.

Each minute is also divided into 60 parts. These parts are
known as seconds. The symbols used to designate an angle, such as
8 degrees; 38 minutes and 40 seconds, would look like this:
8 38’ 40"
A right angle is simply onefourth of a circle. A
right angle is equal to 90 degrees (Figure 2). If an
angle is less than 90 degrees, it is known as an acute
angle. 
Figure 2. A right
angle is 90 degrees. If less than 90 it is acute; if
larger it is obtuse.

If an angle is greater than 90 degrees, it is known as an
obtuse angle. Two angles together, which equal 90 degrees, are
known as complementary angles (Figure 3). 
Figure 3.
Complimentary angles total 90 degrees.

Figure 5. The
clock rotation analogy can be used to describe angles.

Two angles that total 180 degrees would comprise
onehalf of a circle (Figure 5). An angle can be
measured from either direction. In Figure 5 the left
angle could be 45 degrees or 135 degrees. Two angles that total 180
degrees are known as supplementary angles. 
An angle can be an expression of a rotation. Every angle can
be expressed in either a clockwise or a counterclockwise
rotation.
Angle MeasurementPlate
Protractor
The simplest angle measuring device used in the
machining industry is the plate protractor (Figure 6).
The plate protractor is capable of measuring to within
1degree. The plate protractor is especially useful for
layout. 
Figure 6. The flat back
of the plate protractor makes it very good for layout
work.

Angle MeasurementUniversal
Bevel Protractor
The universal bevel protractor picks up where the
blade protractor leaves off. The universal bevel
protractor (Figure 7) is designed for precision
measuring and layout of angles.

Figure 7. The universal
bevel protractor is capable of measuring to within 5
minutes or 1/12 of a degree.

The main component of the bevel protractor is the main scale
. The main scale is graduated into four 90degree components.
The main scale is numbered to read from 0 to 90 degrees and then
back from 90 degrees to 0 (Figure 9).

Figure 9. Degrees can
be read directly off of the main scale, while the
minutes are read on the vernier scale. 
As with other vernier measuring devices, the vernier scale of
the bevel protractor allows the tool to divide each degree into
smaller increments. The vernier scale is divided into 24 spaces, 12
spaces on either side of the zero (see Figure 9).
Each space on the vernier scale is, therefore, onetwelfth of
a degree. Onetwelfth of a degree is equal to 5 minutes. To read
the protractor, note where the zero on the vernier scale lines
up with the degrees on the dial in Figure 10. The degrees are
read directly from the main scale. The zero on the vernier scale
is just pass the 85 degree mark. Now, reading in the same
direction (counterclockwise), count, by five, from zero on
the vernier scale to the lines that match up on the dial (Figure
10).
Figure 10. Always read the vernier in the same direction
that you read the dial.
Add this number of minutes to the number of whole
degrees. The total number of degrees and minutes in Figure 10
would equal 85 degrees and 30 minutes. Look at the measurements
in Figure 11 to get you more accustomed to vernier bevel
protractor reading.
Figure 11. Vernier bevel protractor readings.
Figure 12. When reading
from 90 degrees, make sure to note the positions where
the angle and the supplement are formed.

Any angle can be measured with the vernier bevel
protractor, but you have to be careful to note which
part of a full circle you are measuring. For every
position of the bevel protractor, four angles are formed
(Figure 12). Two of the angles can be read directly on
the main scale and the vernier scale while the other two
are supplemental angles. Keep track of the obtuse and
acute angles and try to read from zero whenever
possible. There is no general rule for use, just keep in
mind that you are adding to 90 degrees to make up the
angle being measured. 
Squares
One of the skills that is continually being tested in
the shop is our ability to machine a surface square to
each other and the ability to check for squareness.
There are many useful tools that can be used to
precisely measure or check for squareness. The most
common perhaps is the solid square (Figure 13). The wide
portion is referred to as the beam and the slender upright
portion is called the blade. The solid square is usually
used to check squareness of surfaces or to square up
parts on a surface plate prior to inspection. 
Figure 13. Solid Square.

Figure 14. The
combination square.

Combination squares (Figure 14) find frequent use
where it is necessary to check for squareness, check
lengths and heights, or for layout work. 
Several methods are available to determine squareness.
After holding the square up to the feature to be
checked, the simplest method is to just look with the
naked eye. For closer work you might use a magnifying
glass or a strong beam of light as an aid to see any
opening that might be there. Even a white sheet of paper
to reflect light might be useful. These methods will
tell if the feature is out of square but not by how much.
The deviation from squareness can be determined by using
feeler stock, paper, or other items of known thickness.
The cylindrical square (Figure 15) is a simple tool
for checking squareness of two planes or a plane and an
edge.
The direct reading cylindrical square indicates
"outofsquareness" of work in units of
.0002" without transfer tools.

Figure 15. Direct
Reading Cylindrical square.

A true cylinder is one with one end lapped perfectly square and the
other end lapped to a fixed angular relation with the sides.
With the angular end down and the base of the cylinder in
contact with the part to be checked, the square is rotated until
light is shut out. Reading up the topmost dotted curve in
contact with the part to the number at the top of the square
shows the outofsquareness of the part in 2, 4, 6, to 12
tenthousandths of an inch. The same reading may be obtained
from two places on the circumference of the square, thus the
instrument is selfchecking. With the squared end down, it may
be used as a master square with one of its perpendicular lines
used for reference.
Square is approximately 2 1/2" in diameter and 6
1/4" high, case hardened with minimum 6 RMS surface
finish. Accuracy within .0001 ".
The universal master square (Figure 16) is a
highprecision square that can be used in any position.
The combination of two broad sides and two knife edged
sides makes it suitable for many applications.

Figure 16. The universal
master square is also known as a toolmakers surface
plate square.

