Triangle geometry

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

A + B + C = 180°

Reads: the three interior angles always spend the same budget, so knowing two angles means you already know the third. Flat plane only — more on that below.

Similarity:   a′/a = b′/b = c′/c = k    (and A′ = A, B′ = B, C′ = C)

Reads: similar triangles share all three angles, and every side of one is the same multiple k of the matching side of the other. Measure any one pair of corresponding sides and k is fixed; every other length follows.

Congruent if:  SSS, SAS, ASA, or AAS match.
Similar if:    two angles match (AA).
Not enough:    SSA (two triangles can fit), AAA (similar, size unknown)

Reads: congruence means identical up to picking the triangle up and setting it down elsewhere, mirror flips included. These letter codes are the complete list of what pins a triangle down and what doesn't.

Scaling:   L′ = k·L      A′ = k²·A      V′ = k³·V

Reads: lengths scale with k, areas with k², volumes and weights with k³. Double a bracket and you get four times the plate area and eight times the mass.

Force triangle:   Fx/F = run/L      Fy/F = rise/L

Reads: for an axial member of length L with horizontal projection run and vertical projection rise, the force components split in the same ratios as the geometry — the force triangle is similar to the space triangle. No trig functions required if you have the dimensions.

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

The half-scale print at the bench. A dimension is missing from the drawing and the engineer who made it retired in 2019. You measure the feature on the print, measure a dimensioned feature to establish k, and back out the missing number. It works because the print and the part are similar figures. The gotcha: a PDF printed "fit to page" is similar to the original in neither direction independently — the office printer applies different scale factors in x and y, and a print that's been through that isn't similar to anything. Check k with two dimensions at right angles before trusting one.

Resolving a diagonal member on the load frame. A brace runs 4 ft over and 3 ft up, so it's 5 ft long, and the load cell in line with it reads 2,000 lb. The horizontal component is 2000·(4/5) = 1600 lb and the vertical is 2000·(3/5) = 1200 lb — the force triangle is similar to the 3-4-5 space triangle, so the components come straight off the drawing dimensions. Every method-of-joints truss solution is this move repeated at each gusset.

Sizing a target you can't reach. A camera is a similar-triangle machine: object height / distance = image height / focal length. Point a camera with a 50 mm lens at a test article 20 m away, count how many millimeters of sensor the article spans, and you have its size without leaving the control room. Shadow methods are the same argument with the sun as the projector — a 1.8 m rod casting a 2.4 m shadow says every object's height is 0.75 of its shadow, so the 32 m shadow off the mast means the mast is 24 m.

The subscale article at the review board. A 1:2.5 model of a tank gets flow-tested and someone scales the results up linearly. The reviewer who catches it is applying the third formula above: wetted area went up by 2.5² = 6.25, volume by 2.5³ ≈ 15.6, and any quantity that rides on area-to-volume ratio does not survive the scale change unchanged. Similarity of geometry never implies similarity of physics — that takes matching the dimensionless groups too.

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

The engine under everything here is the angle-sum constraint. A triangle has six numbers — three sides, three angles — but they are not independent. Two angles determine the third, and once all three angles are set, the triangle's shape is completely determined; only its size is free. That's why AA is enough for similarity, and why one added length measurement upgrades similarity to full knowledge.

The congruence codes are worth actually memorizing, because the failure case bites. SSS, SAS, ASA, and AAS each lock the triangle. SSA does not, and it's the one that shows up in the field: you know two member lengths and an angle that isn't between them. Take sides a = 6, b = 10 with angle A = 30° opposite side a. Two honest triangles satisfy that data — one with B ≈ 56.4°, one with B ≈ 123.6° — and they are different structures. A linkage or a slider mechanism given SSA constraints has two assembly configurations, and a solver or a spreadsheet will happily converge to whichever one it finds first. If your kinematics model jumps branches mid-stroke, this is the reason.

The classic mistake is linear thinking about areas. Similar triangles scale lengths by k and areas by k², and people forget the square under schedule pressure. Doubling a gusset plate "for margin" quadruples its weight; a drawing at 1:10 means an area measured on the drawing is off from reality by a factor of 100, not 10. The square-cube law that limits how big you can build anything is this one line of algebra taken seriously.

Two limits of validity. First, A + B + C = 180° is a flat-plane fact. On a sphere the angles of a triangle sum to more than 180°, and the excess grows with the triangle's area — surveyors closing long geodetic traverses see it and correct for it. Across a machine shop or a test cell, the earth is flat to more precision than your instruments; across a state, it isn't. Second, similar-triangle measurement inherits the conditioning of the triangle you build. A long, skinny triangle — target nearly in line with your baseline, or a shadow at a low sun angle — amplifies small angle errors into large length errors. If the triangle looks degenerate on paper, the answer is degenerate too. Build the fattest triangle the site allows.

One more quiet appearance: the fillet-weld throat. The throat of an equal-leg fillet is the altitude of a 45-45-90 triangle, which is similar to every other 45-45-90 triangle ever drawn, so throat = leg × √2/2 ≈ 0.707 regardless of weld size. That 0.707 on the weld symbol chart is a similarity statement.

History

The oldest engineering measurement story on record is a similar-triangle argument. Thales of Miletus (c. 624–546 BC) is said to have measured the height of an Egyptian pyramid by waiting for the moment his own shadow equaled his own height, then pacing off the pyramid's shadow — a tale that reaches us through Diogenes Laertius and Plutarch, tracing back to Hieronymus of Rhodes ^[1]^[2]. Thales is also credited with a method for finding the distance to ships offshore, which later commentators recognized as an application of triangle congruence, and the ASA congruence result that became Proposition 26 of Euclid's Book I was attributed to him on that basis ^[1]^[2]. How much of this Thales actually proved, nobody can say; the attributions come centuries after the fact ^[1].

The theory nearly died young. Similarity is a statement about ratios of lengths, and the Greek discovery that some lengths — the side and diagonal of a square — have no common measure broke the naive arithmetic of ratios. Eudoxus of Cnidus (408–355 BC) rebuilt proportion on a definition that works whether or not the magnitudes are commensurable, and that theory is generally held to be the substance of Book V of Euclid's Elements ^[3]^[4].

Euclid, around 300 BC, assembled the whole structure: the congruence propositions occupy the first 26 propositions of Book I, and Book VI applies Eudoxus's ratio theory to similar figures ^[3]. The one soft spot sat unrepaired for two millennia — Euclid's argument for SAS relies on picking one triangle up and laying it on the other, a "superposition" move his postulates never license ^[8]. David Hilbert closed the question in his Grundlagen der Geometrie of 1899, which put geometry on a fully formal axiomatic footing and treated a SAS-equivalent statement as a congruence axiom rather than a theorem ^[6]^[7]^[8]. Modern axiom systems for school geometry make the same choice, taking SAS as a postulate outright ^[5].

Related tools

/tools/convert-angle
/tools/weld-throat-stress
/tools/mechanical-advantage-lever

Sources

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