Bearings and azimuths describe the same thing differently
Every line on a survey has a direction. There are two ways to name it — bearing and azimuth — and surveyors switch between them constantly. They're mutually convertible, but mixing conventions inside a single set of computations is the most common entry-level blunder there is.
- Azimuth. A single angle from 0° to 360°, measured clockwise from north. North = 0°, east = 90°, south = 180°, west = 270°. Modern instruments output azimuth directly; modern software prefers it.
- Bearing. A direction expressed as a reference letter (N or S), an angle from 0° to 90°, then a destination letter (E or W). Always written like
N 38° 15' E. Survey plats, deeds, and metes-and-bounds descriptions use bearings exclusively — this is the format you'll read in legal documents.
The quadrant determines the formula
A bearing lives in one of four quadrants — NE, SE, SW, NW. Knowing which quadrant tells you immediately how to convert to and from azimuth. Get the quadrant right and the arithmetic is trivial; get it wrong and every downstream coordinate is wrong.
Working the other direction (bearing → azimuth) is the same formulas in reverse. The interactive figure below lets you drag a compass needle and watch the conversion update in real time — the easiest way to internalise which quadrant you're in.
From direction + distance to point coordinates
Once a line's azimuth and length are known, the coordinate delta is just polar-to-rectangular conversion. Latitudes carry the north–south component; departures carry the east–west. Add them to the starting point's coordinates and you have the ending point.
Sin/cos gotcha: in surveying convention, cos goes with latitude(the north–south axis) and sin goes with departure (the east–west). That's opposite to the x/y convention from math class, where sin is paired with y. The convention exists because azimuth is measured from north, not east.
Drag the compass, predict the bearing
Every surveyor builds a personal shortcut for “which quadrant is this in” — some count clicks from north, some draw the line in their head first. Drag the needle below until you can predict the bearing before you read it off the display.
- 0-90° → N(az)E
- 90-180° → S(180-az)E
- 180-270°→ S(az-180)W
- 270-360°→ N(360-az)W
Try this: sweep through 90°, 180°, 270° and watch the bearing format change at each cardinal. Bearings are always measured from N or S (whichever is closer) toward E or W, and the angle is always 0-90°. Azimuths are simpler for math, bearings are what you'll see on most legal descriptions.
A bearing, a distance, a new point
From point A at (N 5000.00, E 5000.00), run a line bearing S 38° 15' Wfor 425.00 ft. Find the coordinates of the ending point B.
- 1. Convert bearing to azimuth. S 38° 15' W is in the SW quadrant, so azimuth = 180° + 38° 15' =
218° 15'. - 2. Compute the latitude. Δ N = 425.00 × cos(218° 15') = 425.00 × (−0.7848) =
−333.54 ft. - 3. Compute the departure. Δ E = 425.00 × sin(218° 15') = 425.00 × (−0.6197) =
−263.37 ft. - 4. Add to the starting coordinates. NB = 5000.00 + (−333.54) = 4666.46. EB = 5000.00 + (−263.37) = 4736.63.
- 5. Sanity check the quadrant. SW means the new point is south and west of the start. Both Δ N and Δ E came out negative — confirms.
The four intersection types
Once you can run one leg, the next step is finding the point where two lines meet. The exam (and real boundary work) treats four classic intersection problems — which two pieces of information you have determines which formula you use.
| Intersection type | What you know | When to use it |
|---|---|---|
| Bearing-bearing | A point + bearing from station 1, a point + bearing from station 2 | Most common — derive a corner from two found monuments and recovered deed bearings. |
| Distance-distance | A point + distance from station 1, a point + distance from station 2 | Recovering a corner by tape measurement from two reference points (yields two solutions — pick the right one). |
| Bearing-distance | A point + bearing from one station, a point + distance from another | Mixed — one good bearing reference and a taped distance from a second monument. |
| Two distances on a line | A baseline + offset distances | Computing a point along a known line, like a station + offset on a roadway centreline. |
Each of these is a closed-form formula; modern COGO software solves them in a single dialog. Knowing what the formulas do lets you sanity-check the software when it produces a suspicious answer — usually because you fed it the wrong line.
Mistakes coordinate math punishes
- Mixing the sin/cos convention. cos goes with latitude (north–south), sin goes with departure (east–west). Surveying convention; opposite of math class. Burn it in by writing it on the inside cover of your field book.
- Bearing without the quadrant. “38° 15'” alone is meaningless — you need the N/S and E/W letters. Plats that omit them are ambiguous; protest in the boundary survey if you find one.
- Decimal degrees vs DMS in the calculator. cos(218.25) is wrong if you meant 218° 15′ — that's 218.25° in decimal degrees but 218°15′00″ in DMS (which is 218 + 15/60 = 218.25°, so this one happens to match — but 38°15′ = 38.25°, while 38°1.5′ would be 38.025°). Set the calculator mode explicitly every session.
- Forgetting to update the starting point. In a chain of legs, the ending coordinates of leg n become the starting coordinates of leg n+1. Skipping the substitution is the single most common blunder in coordinate-geometry coursework.
How well did it stick?
A quick 5-question check on Coordinate Geometry. See where you stand and what to review.