Angles, Distances, and Trigonometry (FE Civil Surveying)

NCEES FE Civil spec: Surveying, A. Angles, distances, and trigonometry. FE Reference Handbook: Mathematics chapter, Trigonometry; Civil Engineering chapter, Surveying section.

What the FE Civil Spec Covers

Surveying section A asks you to move between the angles and distances a crew measures in the field and the horizontal and vertical quantities that design uses. Expect quick problems: convert a bearing to an azimuth, find a back azimuth, reduce a slope distance to horizontal, or solve an oblique triangle when one side cannot be measured directly. These are usually one or two step problems, so speed and sign discipline matter more than theory.

Key Concepts

Azimuth: angle measured clockwise from north, 0° to 360°. The back azimuth differs by 180°.
Bearing: acute angle (0° to 90°) from north or south toward east or west, written like N 30° E or S 35° E. Quadrant rules: NE bearing = azimuth; SE: azimuth = 180° minus bearing; SW: azimuth = 180° plus bearing; NW: azimuth = 360° minus bearing.
Vertical angle $α$ is measured up or down from horizontal; the zenith angle $z = 9 0^{\circ} - α$ is measured down from straight up. Total stations usually report zenith angles.
Interior angles of a closed polygon with n sides sum to $(n - 2) 18 0^{\circ}$ , which is the first check on any closed traverse.

Core Formulas

Slope distance S reduced with a vertical angle:

H = S cos α V = S sin α

The same with a zenith angle:

H = S sin z V = S cos z

Elevation of a sighted point B from an instrument over point A (instrument height hi, rod or target height r):

Elev_{B} = Elev_{A} + hi + V - r

For triangles that are not right triangles:

\frac{a}{sin A} = \frac{b}{sin B} = \frac{c}{sin C} c^{2} = a^{2} + b^{2} - 2 ab cos C

Where It Is in the FE Reference Handbook

The trig identities and the laws of sines and cosines are in the Mathematics chapter under Trigonometry. The sign convention for latitudes and departures is in the Civil Engineering chapter, Surveying section. Bearing to azimuth conversion is not spelled out, so memorize the quadrant rules. See the FE handbook section guide.

Worked Example

A total station is set over point A (elevation 152.40 m) with an instrument height of 1.55 m. It sights a prism on a 1.80 m pole at point B and reads a slope distance of 245.60 m and a zenith angle of 84°30'00". Find the horizontal distance AB and the elevation of B.

Step 1, vertical angle. 84°30' = 84.5°, so the line is inclined upward by

α = 9 0^{\circ} - 84. 5^{\circ} = 5. 5^{\circ}

Step 2, horizontal and vertical components.

H = 245.60 cos 5. 5^{\circ} = 244.47 m

V = 245.60 sin 5. 5^{\circ} = 23.54 m

Step 3, elevation of B. Add the instrument height, add the rise, subtract the pole height:

Elev_{B} = 152.40 + 1.55 + 23.54 - 1.80 = 175.69 m

The horizontal distance is 244.47 m and B is at 175.69 m. Note that the horizontal correction (1.13 m) is large even at a modest 5.5° slope; ignoring it would fail any closure check.

Exam Strategy

Most section A questions are answerable in under two minutes if you sketch first. Draw a north arrow, place the line in its quadrant, and label the angle you were given before touching the calculator. That sketch catches nearly every bearing and azimuth error.

For oblique triangles, decide which law to use from what you know. Two angles and a side, or two sides and an angle opposite one of them: law of sines. Two sides and the included angle, or all three sides: law of cosines. A classic case is an inaccessible distance across a river: measure a baseline on your side and the angles to a point on the far bank, find the third angle from the 180° sum, and apply the law of sines.

Watch the answer choices. If two choices differ by roughly the cosine factor (for example 244.5 m versus 245.6 m), the exam is checking whether you reduced the slope distance at all.

Common Mistakes

Using cosine with a zenith angle as if it were a vertical angle (this gives V, not H).
Typing 84.30 for 84°30'. Thirty minutes is 0.5°, not 0.3°.
Forgetting the instrument height or pole height when carrying elevations by trig leveling.
Getting the SW and NW quadrant conversions backwards. Sketch the line before converting.
Back azimuth: add 180° if the azimuth is under 180°, otherwise subtract 180°. The result must stay between 0° and 360°.

Practice Problems

Try these from the free PolyCalc FE Civil problem bank. Each one has a full worked solution.

The bearing N30°E corresponds to what azimuth, measured clockwise from north? See the worked solution or practice the Surveying deck
The bearing S35°E corresponds to an azimuth (from north) of: See the worked solution or practice the Surveying deck
The back azimuth of 287° is: See the worked solution or practice the Surveying deck
A slope distance of 100 m is measured at 10° above the horizontal. What is the approximate horizontal distance? See the worked solution or practice the Surveying deck
A slope distance of 150.00 m is measured at a vertical angle of 8°. The horizontal distance is most nearly: See the worked solution or practice the Surveying deck

Related Calculators and Examples

Frequently Asked Questions

What is the difference between a bearing and an azimuth?

An azimuth is a single angle from 0 to 360 degrees measured clockwise from north. A bearing is an angle from 0 to 90 degrees measured from north or south toward east or west, such as N 30 E or S 35 E. S 35 E equals an azimuth of 145 degrees.

Should my calculator be in degrees or radians for FE surveying problems?

Degrees. Surveying angles are given in degrees, minutes and seconds. Convert minutes and seconds to decimal degrees (divide minutes by 60 and seconds by 3600) before using sine or cosine.

How do I get horizontal distance from a zenith angle?

A zenith angle is measured down from vertical, so horizontal distance equals slope distance times the sine of the zenith angle and the vertical difference equals slope distance times its cosine. With a vertical angle measured from horizontal, swap sine and cosine.

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