Triangle solving and triangulation

The formula (the core equation(s), each with a one-line reading of what it says)

A + B + C = 180°

Reads: two angles buy you the third for free, and the sum is your closure check — if your three measured angles don't add to 180° (plus spherical excess on long baselines), the difference is your error, laid out in plain sight.

a / sin(A) = b / sin(B) = c / sin(C)        (law of sines)

Reads: sides and the sines of their opposite angles keep one fixed ratio, so a known side-angle pair unlocks every other pair. This is the workhorse for the angle-heavy cases.

c² = a² + b² − 2·a·b·cos(C)                 (law of cosines)

Reads: Pythagoras with a correction term for the angle not being 90°. This is the workhorse for the side-heavy cases — two sides and the angle between them (a = 8, b = 5, C = 60° gives c = 7 exactly), or three sides and no angles at all.

d_A = b · sin(β) / sin(α + β)               (intersection from a baseline)

Reads: stand at both ends of a baseline of length b, measure the angle to the target from each end (α, β), and the distance from station A to the target drops out. The sin(α + β) in the denominator is the whole conditioning story — more on that below.

Z = f·B / d                                 (stereo depth from disparity)

Reads: the same triangle in camera clothing — two cameras a baseline B apart, focal length f in pixels, and a feature that lands d pixels apart in the two images sits at depth Z. Small disparity means far away, and far away means fragile.

Where you meet it (2-4 concrete engineering situations, specific: bench, test stand, review board)

Laying out a test stand you can't tape across. Anchor-bolt patterns, thrust-stand centerlines, and instrument piers get set by total station, and every shot the instrument can't reach directly becomes an intersection or resection problem. Two setups 200 m apart each sight a monument across the flame trench; angles of 55° and 63° off the baseline put the monument 200·sin(63°)/sin(62°) = 201.83 m from the first station and 185.55 m from the second. Nobody stretched a tape over the trench — but note that with only two setups there is no closure check: the 62° is forced by the other two angles, not measured, so the 180° sum is an identity, not a receipt. A third setup sighting the same monument turns that identity into a real check, which is why layout crews shoot critical monuments from three stations.

Direction finding. Two receive sites on a known baseline each measure an angle of arrival to an emitter, and the fix is the intersection formula verbatim. A 10 km baseline with a 20° angle at the emitter puts it about 25 km out — workable. The same emitter sitting near the baseline's extension collapses the triangle to a sliver, and the fix ellipse stretches to uselessness. The RF crowd calls this geometry problem out in DOP terms; the surveyor calls it a skinny triangle; it is the same disease.

The stereo-vision bench. Depth cameras, photogrammetry rigs, and stereo DIC on a structures test all run Z = f·B/d. With f = 2000 px and a 120 mm baseline, a target at 10 m produces 24 px of disparity — and a half-pixel matching error moves the computed depth by about 0.21 m. The same half pixel at 2 m range moves it 8 mm. Depth error grows as Z²·Δd/(f·B), which is why the motion-capture vendor's accuracy spec quietly assumed you'd read the fine print about working volume.

The review board. When the tracking or metrology plan comes up, someone will ask why the camera baseline or the DF site separation is what it is. The honest answer is always a triangle-conditioning argument: intersection angles between roughly 30° and 150°, or the position error budget doesn't close.

How it works (the real substance — behavior, gotchas, limits of validity, the mistake people make)

A triangle has six parts — three sides, three angles — and three independent measurements determine it, with one nonnegotiable condition: at least one measurement must be a side. Three angles (AAA) fix only the shape; every scaled copy has the same angles, so there is no size without a length somewhere. That is why every triangulation network in history starts with somebody taping, chaining, or laser-ranging one baseline with obsessive care: the baseline is the only place absolute scale enters, and every distance in the network inherits its error.

The five solvable cases, and the right tool for each:

ASA / AAS — angle sum gives the third angle, law of sines gives the sides. Unambiguous. This is the surveyor's intersection case.
SAS — law of cosines for the third side, then law of sines (for the smaller remaining angle) to finish.
SSS — law of cosines rearranged, cos(C) = (a² + b² − c²)/(2ab), three times or twice plus the angle sum. Also unambiguous, and acos correctly reports obtuse angles, which asin never does.
SSA — the ambiguous case: zero, one, or two triangles fit the data, and when there are two, both are physically real (the open and crossed configurations of a linkage, the two sides of a DF baseline). The math is not misbehaving; your measurements genuinely don't distinguish them, and only outside knowledge picks the branch.

Triangulation is triangle solving made industrial. Measure one baseline, then measure angles from station to station, chaining triangles across the countryside; each solved triangle's sides become baselines for the next. Angles were the smart currency for four centuries because a theodolite reads arc-seconds cheaply while distance measurement stayed slow and painful until laser rangefinders arrived. Two habits made the chains trustworthy: keep every intersection angle well conditioned, and check every triangle's angle sum. The 1/sin(α+β) factor in the intersection formula says it directly — at a 90° intersection angle the geometry is as good as it gets, while at 5° the same bearing errors hit the computed position about 11.5 times harder.

The mistakes people make:

Calling trilateration "triangulation." GPS and UWB tags measure ranges, not angles — that's trilateration (spheres intersecting), a different problem with different failure geometry. The words get swapped in half the briefings you'll sit through. Angles-in, triangulation; distances-in, trilateration; time-differences-in, multilateration.
Trusting asin in the side-heavy cases. After a law-of-cosines start, finish with the law of sines on the smaller angle — anything opposite a shorter side is guaranteed acute, so the calculator's principal value is safe. Solve for the big angle by subtraction.
Ignoring conditioning until the data is taken. A skinny triangle doesn't announce itself in the residuals of a single fix; it announces itself when the "fixed" point wanders 40 m between sessions. Geometry is a design decision, made before anyone measures anything.
Forgetting the plane is an approximation. Over a machine shop or a launch pad, flat trig is exact for any purpose you have. Over tens of kilometers the triangle lives on the ellipsoid, angles sum to more than 180° (spherical excess), and geodetic reductions take over. The crossover is farther out than intuition says, but it exists, and long-baseline instrumentation ranges cross it.

History (who derived it and when, told as a short story with inline [n] citations)

The triangle-solving toolkit — sine law, cosine law, the case analysis — was assembled by medieval Islamic mathematicians and packaged for Europe by Regiomontanus in the 1460s. Triangulation as a method is younger and has a birth certificate: in 1533 Gemma Frisius, a physician-mathematician in Leuven, appended a sixteen-page booklet, Libellus de locorum describendorum ratione, to an edition of Apianus's Cosmographia, proposing for the first time that a country could be mapped by measuring one baseline and then only angles — sighting each town from two known points and intersecting the rays ^[1]^[2]. His worked illustration used Brussels and Antwerp as the baseline. It was a paper method; Frisius never ran the survey himself.

Willebrord Snellius did. In 1615 he chained a network of triangles — fourteen stations, mostly church towers, 53 triangle measurements — from Alkmaar to Bergen op Zoom, about 130 km on roughly the same meridian, reading angles with a large quadrant and anchoring the scale with a measured baseline. He published in Eratosthenes Batavus (1617), styling himself the Dutch Eratosthenes, and got the Earth's circumference within a few percent ^[3]^[4]. That book is the founding document of geodesy: one careful length, disciplined angles, and the law of sines carrying scale across a whole province.

The method's monument is the Great Trigonometrical Survey of India, begun by William Lambton on 10 April 1802 with a baseline near Madras and pushed up the subcontinent by George Everest and Andrew Waugh over nearly seventy years ^[5]. At its northern edge, between November 1849 and January 1850, observers took theodolite readings of a Himalayan summit logged as Peak XV from six stations, each 108 to 118 miles from the mountain — Nepal was closed, so the triangle sides were enormous and the refraction corrections brutal. The six computed heights ranged from 28,990 to 29,026 ft ^[6]. In 1852 Radhanath Sikdar, the survey's chief computer in Calcutta, worked the numbers through and concluded Peak XV was the highest mountain on Earth; Waugh announced the height as 29,002 ft in March 1856 ^[7]^[6]. The modern surveyed value is 29,032 ft — the 1850s triangulation, shot from over 100 miles away through unmodeled refraction, landed within about 0.1%.

Related tools (bullet list of HE calculator slugs that use or neighbor this topic, as /tools/ links)

/tools/convert-angle — degrees, radians, and mils squared away before any bearing hits a sine
/tools/radar-horizon — whether your two stations can see the target at all is a prior question to the triangle
/tools/fresnel-zone — path geometry over a surveyed baseline, same endpoints, different question
/tools/doppler-shift — what the moving emitter does to your DF problem while you're still solving the triangle

Sources

Written by HE in our own words from the cited sources — engineering judgment included, your stamp still required. All entries →