Theodolite
The theodolite measures horizontal and vertical angles. Learn instrument parts and temporary adjustments, face-left/face-right observation, bearings (whole-circle and reduced), traverse computation (latitudes, departures), closing error and Bowditch adjustment, and trigonometric levelling.
Where tapes measure distances, the theodolite measures angles — to seconds of arc. Combined with distances, angles fix positions: traverses around a site, control networks, tunnel alignments and tall-building verticality.
Traverse computation — turning angles and lengths into coordinates, then adjusting small errors — is a core surveying skill.
An analogy
Walking around a field noting each turn and step count: if you end up where you started, your notes are consistent; the small gap you're left with is the closing error.
Instrument and set-up
Temporary adjustments: centre over the station (plumb/optical plummet), level with plate bubbles, focus eyepiece and object (remove parallax). Observe each angle on face left and face right and average — this cancels most instrumental errors.
Bearings
Whole-circle bearings run 0–360° clockwise from north; reduced (quadrantal) bearings like N30°E are used in some fields. Forward and back bearings differ by 180°. Angles between lines convert to bearings along a traverse.
Traverse computation
Closed traverses return to the start (or another known point). Sum latitudes and departures; the residual is the misclosure. Precision of 1:5000–1:20 000 is typical for engineering surveys. The Bowditch (compass) rule distributes corrections in proportion to leg length; the transit rule in proportion to latitudes/departures.
Heights, control and modern practice
Trigonometric levelling finds heights from vertical angles and distances (with curvature/refraction for long lines). Electronic theodolites and total stations now dominate; least-squares adjustment of control networks gives rigorous coordinates and error ellipses.
Formulas
- Latitude & departure:
ΔN = L cos θ, ΔE = L sin θ - Closing error:
e = √(ΣΔN² + ΣΔE²), precision = 1 : (ΣL/e) - Bowditch correction:
δN_i = −ΣΔN · L_i/ΣL, δE_i = −ΣΔE · L_i/ΣL - Interior angles check:
Σ = (n − 2) × 180° - Trigonometric levelling:
Δh = D tan α + h_i − h_t
Worked example: Closed traverse misclosure
Given: AB 100 m @ 30°, BC 120 m @ 120°; CD 90 m @ 210°, DA 120.40 m @ 295°15′
Find: Misclosure and precision.
- ΣΔN = 86.603 − 60.000 − 77.942 + 51.358 = +0.019 m
- ΣΔE = 50.000 + 103.923 − 45.000 − 108.896 = +0.027 m
- e = √(0.019² + 0.027²) = 0.033 m
- Precision = 430.4/0.033 ≈ 1 : 13 000
Answer: e ≈ 33 mm, 1:13 000 ✓
Worked example: Trigonometric height
Given: Horizontal distance 200 m, vertical angle +8°; Instrument height 1.5 m, target height 2.0 m
Find: Height difference.
- Δh = 200 tan 8° + 1.5 − 2.0 = 28.108 − 0.5 = 27.61 m
Answer: ≈ 27.6 m
Common mistakes
- Wrong: Observing only on one face. — Right: Average face left and face right to remove instrument errors.
- Wrong: Distributing misclosure equally regardless of leg length. — Right: Use Bowditch (proportional to length) or least squares.
Applications
- Control surveys: Traverses establishing site control for construction.
- Tunnels & shafts: Precise alignment transfer underground.
- Boundary surveys: Legal parcel boundaries from bearings and distances.
Key takeaways
- ΔN = L cos θ, ΔE = L sin θ.
- Misclosure → precision 1:(ΣL/e).
- Adjust with Bowditch or least squares.
- Observe both faces; check angle sums.
Going further
- Least-squares adjustment: Rigorous network adjustment with weights and error ellipses.
- Gyro-theodolites: Determine true north underground for tunnel orientation.
- Resection & intersection: Fix unknown points from angles to known points.