Engineer Exam Lab Start free trial
Menu

Independent study resource, not affiliated with NCEES, Pearson VUE, or any state licensing board. You register for the exams and pay their fees only at ncees.org. This site collects no logins and offers no downloads.

10 FE practice problems: surveying: areas, coordinates, earthwork, with solutions

By Dr. Mohamed Elansary, PhD Environmental EngineeringUpdated

These ten original problems practice surveying: areas, coordinates, earthwork, a topic from the FE Civil exam specification, using the Civil Engineering, Earthwork Formulas part of the FE Reference Handbook 10.6. Each problem gives the situation and the values with units. Work it with the handbook PDF open and commit to an answer before you open the solution. Every solution shows the handbook page, the equation, the substitution with units, a size check, and the mistake behind each wrong option, so a wrong pick tells you exactly what to fix.

How to use these problems

Give each problem an honest attempt before you open the solution: write the given values with units, find the equation in the FE Reference Handbook PDF, and commit to an answer. Then compare line by line. If you picked a wrong option, read the note for that option; each one names the mistake that produces it.

The problems

Problem 1 · FE Civil, Surveying: areas, coordinates, earthwork

Two cross sections of a road cut, 100 ft apart, have end areas of 293 ft² and 162 ft². Find the volume of excavation between them in yd³ by the average end area method.

Enter your answer in yd³, 3 significant figures.

Handbook: Civil Engineering, Earthwork Formulas, FE Reference Handbook 10.6

Show the worked solution
Answer
843 yd³
Given
A1 = 293 ft², A2 = 162 ft², L = 100 ft
Find
earthwork volume (yd³)
Handbook
Civil Engineering, Earthwork Formulas, Average End Area Formula, page 315
Equation
V = L(A1 + A2)/2
Substitute
  1. V = L(A1 + A2)/2 = 100(293 + 162)/2 = 22,750 ft³
  2. V = 22,750 ft³ ÷ 27 ft³/yd³ = 843 yd³
Result
843 yd³, 3 significant figures
Check
the volume lies between L × (smaller area) and L × (larger area) in yd³: 600 to 1,090.
Common wrong answers
  • 22,800 yd³: left the volume in ft³ (no ÷ 27)
  • 2,530 yd³: divided by 9 (ft² per yd²) instead of 27 (ft³ per yd³)
  • 1,690 yd³: added the two areas without averaging them

Problem 2 · FE Civil, Surveying: areas, coordinates, earthwork

Two cross sections of a road cut, 100 ft apart, have end areas of 464 ft² and 386 ft². Find the volume of excavation between them in yd³ by the average end area method.

Enter your answer in yd³, 3 significant figures.

Handbook: Civil Engineering, Earthwork Formulas, FE Reference Handbook 10.6

Show the worked solution
Answer
1,570 yd³
Given
A1 = 464 ft², A2 = 386 ft², L = 100 ft
Find
earthwork volume (yd³)
Handbook
Civil Engineering, Earthwork Formulas, Average End Area Formula, page 315
Equation
V = L(A1 + A2)/2
Substitute
  1. V = L(A1 + A2)/2 = 100(464 + 386)/2 = 42,500 ft³
  2. V = 42,500 ft³ ÷ 27 ft³/yd³ = 1,570 yd³
Result
1,570 yd³, 3 significant figures
Check
the volume lies between L × (smaller area) and L × (larger area) in yd³: 1,430 to 1,720.
Common wrong answers
  • 42,500 yd³: left the volume in ft³ (no ÷ 27)
  • 4,720 yd³: divided by 9 (ft² per yd²) instead of 27 (ft³ per yd³)
  • 3,150 yd³: added the two areas without averaging them

Problem 3 · FE Civil, Surveying: areas, coordinates, earthwork

Two cross sections of a road cut, 100 ft apart, have end areas of 158 ft² and 389 ft². Find the volume of excavation between them in yd³ by the average end area method.

Enter your answer in yd³, 3 significant figures.

Handbook: Civil Engineering, Earthwork Formulas, FE Reference Handbook 10.6

Show the worked solution
Answer
1,010 yd³
Given
A1 = 158 ft², A2 = 389 ft², L = 100 ft
Find
earthwork volume (yd³)
Handbook
Civil Engineering, Earthwork Formulas, Average End Area Formula, page 315
Equation
V = L(A1 + A2)/2
Substitute
  1. V = L(A1 + A2)/2 = 100(158 + 389)/2 = 27,350 ft³
  2. V = 27,350 ft³ ÷ 27 ft³/yd³ = 1,010 yd³
Result
1,010 yd³, 3 significant figures
Check
the volume lies between L × (smaller area) and L × (larger area) in yd³: 585 to 1,440.
Common wrong answers
  • 2,030 yd³: added the two areas without averaging them
  • 3,040 yd³: divided by 9 (ft² per yd²) instead of 27 (ft³ per yd³)
  • 27,400 yd³: left the volume in ft³ (no ÷ 27)

Problem 4 · FE Civil, Surveying: areas, coordinates, earthwork

A traverse course has a bearing of N 39°15′ E and a length of 274.6 ft. Find the length of its departure (east-west component) in ft.

Enter your answer in ft, 3 significant figures.

Handbook: Civil Engineering, Latitudes and Departures, FE Reference Handbook 10.6

Show the worked solution
Answer
174 ft
Given
L = 274.6 ft, deg = 39°, min = 15′
Find
departure (ft)
Handbook
Civil Engineering, Surveying, Latitudes and Departures, page 315
Equation
Latitude = L cos θ, Departure = L sin θ (θ = bearing angle from the meridian)
Substitute
  1. θ = 39° + 15/60 = 39.25°
  2. Departure = L sin θ = 274.6 × sin 39.25° = 174 ft (east (+))
Result
174 ft, 3 significant figures
Check
latitude L cos θ = 213 ft; latitude² + departure² = L² (274.6²).
Common wrong answers
  • 224 ft: used the tangent of the bearing angle
  • 213 ft: is the latitude (L cos θ), not the departure

Problem 5 · FE Civil, Surveying: areas, coordinates, earthwork

Two cross sections of a road cut, 100 ft apart, have end areas of 291 ft² and 219 ft². Find the volume of excavation between them in yd³ by the average end area method.

Enter your answer in yd³, 3 significant figures.

Handbook: Civil Engineering, Earthwork Formulas, FE Reference Handbook 10.6

Show the worked solution
Answer
944 yd³
Given
A1 = 291 ft², A2 = 219 ft², L = 100 ft
Find
earthwork volume (yd³)
Handbook
Civil Engineering, Earthwork Formulas, Average End Area Formula, page 315
Equation
V = L(A1 + A2)/2
Substitute
  1. V = L(A1 + A2)/2 = 100(291 + 219)/2 = 25,500 ft³
  2. V = 25,500 ft³ ÷ 27 ft³/yd³ = 944 yd³
Result
944 yd³, 3 significant figures
Check
the volume lies between L × (smaller area) and L × (larger area) in yd³: 811 to 1,080.
Common wrong answers
  • 1,890 yd³: added the two areas without averaging them
  • 25,500 yd³: left the volume in ft³ (no ÷ 27)
  • 2,830 yd³: divided by 9 (ft² per yd²) instead of 27 (ft³ per yd³)

Problem 6 · FE Civil, Surveying: areas, coordinates, earthwork

A traverse course has a bearing of S 9°30′ W and a length of 536.0 m. Find the length of its departure (east-west component) in m.

Enter your answer in m, 3 significant figures.

Handbook: Civil Engineering, Latitudes and Departures, FE Reference Handbook 10.6

Show the worked solution
Answer
88.5 m
Given
L = 536.0 m, deg = 9°, min = 30′
Find
departure (m)
Handbook
Civil Engineering, Surveying, Latitudes and Departures, page 315
Equation
Latitude = L cos θ, Departure = L sin θ (θ = bearing angle from the meridian)
Substitute
  1. θ = 9° + 30/60 = 9.500°
  2. Departure = L sin θ = 536.0 × sin 9.500° = 88.5 m (west (-))
Result
88.5 m, 3 significant figures
Check
latitude L cos θ = 529 m; latitude² + departure² = L² (536.0²).
Common wrong answers
  • 86.6 m: read 9°30′ as 9.30°
  • 529 m: is the latitude (L cos θ), not the departure

Problem 7 · FE Civil, Surveying: areas, coordinates, earthwork

Two cross sections of a road cut, 100 ft apart, have end areas of 747 ft² and 794 ft². Find the volume of excavation between them in yd³ by the average end area method.

Enter your answer in yd³, 3 significant figures.

Handbook: Civil Engineering, Earthwork Formulas, FE Reference Handbook 10.6

Show the worked solution
Answer
2,850 yd³
Given
A1 = 747 ft², A2 = 794 ft², L = 100 ft
Find
earthwork volume (yd³)
Handbook
Civil Engineering, Earthwork Formulas, Average End Area Formula, page 315
Equation
V = L(A1 + A2)/2
Substitute
  1. V = L(A1 + A2)/2 = 100(747 + 794)/2 = 77,050 ft³
  2. V = 77,050 ft³ ÷ 27 ft³/yd³ = 2,850 yd³
Result
2,850 yd³, 3 significant figures
Check
the volume lies between L × (smaller area) and L × (larger area) in yd³: 2,770 to 2,940.
Common wrong answers
  • 8,560 yd³: divided by 9 (ft² per yd²) instead of 27 (ft³ per yd³)
  • 77,100 yd³: left the volume in ft³ (no ÷ 27)
  • 5,710 yd³: added the two areas without averaging them

Problem 8 · FE Civil, Surveying: areas, coordinates, earthwork

A traverse course has a bearing of N 42°45′ W and a length of 446.9 m. Find the length of its departure (east-west component) in m.

Enter your answer in m, 3 significant figures.

Handbook: Civil Engineering, Latitudes and Departures, FE Reference Handbook 10.6

Show the worked solution
Answer
303 m
Given
L = 446.9 m, deg = 42°, min = 45′
Find
departure (m)
Handbook
Civil Engineering, Surveying, Latitudes and Departures, page 315
Equation
Latitude = L cos θ, Departure = L sin θ (θ = bearing angle from the meridian)
Substitute
  1. θ = 42° + 45/60 = 42.75°
  2. Departure = L sin θ = 446.9 × sin 42.75° = 303 m (west (-))
Result
303 m, 3 significant figures
Check
latitude L cos θ = 328 m; latitude² + departure² = L² (446.9²).
Common wrong answers
  • 413 m: used the tangent of the bearing angle
  • 328 m: is the latitude (L cos θ), not the departure

Problem 9 · FE Civil, Surveying: areas, coordinates, earthwork

A traverse course has a bearing of S 28°50′ E and a length of 681.2 ft. Find the length of its departure (east-west component) in ft.

Enter your answer in ft, 3 significant figures.

Handbook: Civil Engineering, Latitudes and Departures, FE Reference Handbook 10.6

Show the worked solution
Answer
329 ft
Given
L = 681.2 ft, deg = 28°, min = 50′
Find
departure (ft)
Handbook
Civil Engineering, Surveying, Latitudes and Departures, page 315
Equation
Latitude = L cos θ, Departure = L sin θ (θ = bearing angle from the meridian)
Substitute
  1. θ = 28° + 50/60 = 28.83°
  2. Departure = L sin θ = 681.2 × sin 28.83° = 329 ft (east (+))
Result
329 ft, 3 significant figures
Check
latitude L cos θ = 597 ft; latitude² + departure² = L² (681.2²).
Common wrong answers
  • 597 ft: is the latitude (L cos θ), not the departure
  • 375 ft: used the tangent of the bearing angle

Problem 10 · FE Civil, Surveying: areas, coordinates, earthwork

A parcel's corners, in order around the boundary, have coordinates (X, Y) in m: A (1,056, 1,042), B (889, 1,192), C (955, 819), D (1,046, 784). Find the area of the parcel in m².

Enter your answer in m², 3 significant figures.

Handbook: Civil Engineering, Area by Coordinates, FE Reference Handbook 10.6

Show the worked solution
Answer
38,100 m²
Given
X = 1,056, 889, 955, 1,046, Y = 1,042, 1,192, 819, 784
Find
area (m²)
Handbook
Civil Engineering, Surveying, Area by Coordinates, page 316
Equation
Area = [XA(YB - YN) + XB(YC - YA) + ... + XN(YA - YN-1)]/2
Substitute
  1. Area = |Σ X_i (Y_(i+1) - Y_(i-1))|/2 = |1,056(1,192 - 784) + 889(819 - 1,042) + 955(784 - 1,192) + 1,046(1,042 - 819)|/2
  2. = |76,219|/2 = 38,100 m²
Result
38,100 m², 3 significant figures
Check
going around the boundary in the other direction only changes the sign of the sum, not the area.
Common wrong answers
  • 19,100 m²: divided by 2 twice
  • 76,200 m²: forgot to divide the coordinate sum by 2

A new problem posts every day inside the lab, with the full worked solution the same evening.

Frequently asked questions

Where is surveying: areas, coordinates, earthwork in the FE Reference Handbook?

Look in the Civil Engineering, Earthwork Formulas part of FE Reference Handbook 10.6. Each solution gives the exact page, so you can practice finding it in the PDF the way you will on exam day.

Are these real FE exam questions?

No. They are original problems generated from the handbook formulas and checked by code. Real exam questions are confidential, and sharing them breaks the NCEES agreement every examinee accepts.

How do I check my answer before opening the solution?

Check the units of your result and whether its size makes sense for the situation. Each solution ends with the same kind of check, so you can compare your habit with ours.

Sources

  1. NCEES FE Civil CBT exam specifications (PDF). Retrieved October 3, 2026.
  2. NCEES FE Reference Handbook 10.6 (free PDF in MyNCEES). Retrieved October 3, 2026.