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Inspection Methods - Angular Measurement
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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 one-fourth 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 one-half 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 counter-clockwise rotation.

Angle Measurement-Plate 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 1-degree. 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 Measurement-Universal 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 universal bevel protractor is capable of measuring obtuse angles as well as acute angles when accompanied with the correct attachments. Look at Figure 8 below to give you an idea as to the uses of the universal bevel protractor.

Measuring Acute Angles

Measuring Obtuse Angles

Using a protractor with a vernier height gage.

Figure 8. Measuring applications for the universal bevel protractor.

The main component of the bevel protractor is the main scale . The main scale is graduated into four 90-degree 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, one-twelfth of a degree. One-twelfth 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 (counter-clockwise), 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.


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 "out-of-squareness" 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 out-of-squareness of the part in 2, 4, 6, to 12 ten-thousandths of an inch. The same reading may be obtained from two places on the circumference of the square, thus the instrument is self-checking. 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 high-precision 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 tool-makers surface plate square.

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