Slope Distance to Horizontal Distance practice problems for the FE Civil exam

An electronic distance meter measures the straight line between two instruments, which on sloping ground is longer than the horizontal distance that appears on a plan. Reducing one to the other is a single trigonometric step, and the FE Civil exam tests whether you use the right angle in it. When the angle is a vertical angle measured up or down from horizontal, the horizontal distance is the slope distance times the cosine of that angle. When the instrument reports a zenith angle, measured down from straight up, the horizontal distance is the slope distance times the sine of the zenith angle instead. The two conventions differ by ninety degrees, and reading one as the other is the most common way to lose these items. A related version gives the elevation difference rather than an angle: then the horizontal distance follows from the Pythagorean theorem, as the square root of the slope distance squared minus the elevation difference squared. Two sanity checks cost nothing. The horizontal distance is always shorter than the slope distance, so an answer larger than the measurement is wrong. And on gentle grades the two differ by very little, so a large reduction on a shallow slope signals a mode or convention error rather than a real result.

Key formulas

From a vertical angle measured from horizontal: H = S * cos(theta)
From a zenith angle measured from vertical: H = S * sin(z)
Elevation difference from a vertical angle: delta_elev = S * sin(theta)
From slope distance and elevation difference: H = sqrt(S^2 - delta_elev^2)
Grade from the reduced distances: grade = delta_elev / H

Common mistakes

Applying cosine to a zenith angle, which is measured from vertical and needs sine.
Returning a horizontal distance longer than the measured slope distance.
Leaving the calculator in radians so that cos(4) is read as an angle in radians.
Using the slope distance directly in a grade calculation instead of the reduced horizontal distance.

8 worked practice problems

Commit to an answer before opening the solution. Each problem shows the governing model, every substitution, a unit check, and why each wrong choice is wrong.

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Background reading

Original question written by FE Exam AI Prep from the public FE Civil exam specification. Before release, a solver model re-derived the answer twice from the question alone, without its answer key, and a critic model checked the result. No licensed engineer reviews these questions. It is not an NCEES question and does not come from any NCEES practice exam. FE Exam AI Prep is an independent study tool and is not affiliated with, endorsed by, or sponsored by NCEES.

Slope Distance to Horizontal Distance practice problems for the FE Civil exam

Key formulas

From a vertical angle measured from horizontal: H = S * cos(theta)
From a zenith angle measured from vertical: H = S * sin(z)
Elevation difference from a vertical angle: delta_elev = S * sin(theta)
From slope distance and elevation difference: H = sqrt(S^2 - delta_elev^2)
Grade from the reduced distances: grade = delta_elev / H

Common mistakes

Applying cosine to a zenith angle, which is measured from vertical and needs sine.
Returning a horizontal distance longer than the measured slope distance.
Leaving the calculator in radians so that cos(4) is read as an angle in radians.
Using the slope distance directly in a grade calculation instead of the reduced horizontal distance.

8 worked practice problems

Commit to an answer before opening the solution. Each problem shows the governing model, every substitution, a unit check, and why each wrong choice is wrong.