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In trigonometry and geometry, 'triangulation' is the process of finding coordinates and distance to a point by calculating the length of one side of a triangle, given measurements of angles and sides of the triangle formed by that point and two other known reference points, using the law of sines.
(In the figure at right, the third angle of the triangle (call it ''θ'') is known to be 180-α-β, since the sum of the three angles in any triangle is known to be 180 degrees. The opposite-side for this (the third) angle is ''l'', which is a known distance. Since, by the law of sines, the ratio ''sin''(θ)/''l'' is equal to that same ratio for the other two angles ''α'' and ''β'', the lengths of any of the remaining two sides can be computed by algebra. Given either of these lengths, sine and cosine can be used to calculate the offsets in both the north/south and east/west axes from the corresponding observation point to the unknown point, thereby giving its final coordinates.)
Some identities often used (valid only in flat or euclidean geometry):

★ The sum of the angles of a triangle is π, rad or 180 degrees.

★ The law of sines

★ The law of cosines

★ The Pythagorean theorem

Contents

Calculation

See also

Calculation

★ α, β and distance AB are already known

★ C can be calculated by using the distance RC or MC:

★ 'RC:' Position of C can be calculated using law of sines and law of cosines
:
:
Now we can calculate AC and BC
:
:
Last step is to calculate RC via
:
:or
:

★ 'MC' can be calculated using the Pythagorean theorem
:
ight)-left(BC cdot coseta
ight)
:$MC=sqrt\{MR^2+RC^2\}$
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Triangulation is used for many purposes, including surveying, navigation, metrology, astrometry, binocular vision, model rocketry and gun direction of weapons.
Many of these surveying problems involve the solution of large meshes of triangles, with hundreds or even thousands of observations. Complex triangulation problems involving real-world observations with errors require the solution of large systems of simultaneous equations to generate solutions.
Famous uses of triangulation have included the retriangulation of Great Britain.

See also

★ GSM localization

★ Multilateration, where a point is calculated using the time-difference-of-arrival between other known points

★ Parallax

★ Resection

★ SOCET SET

★ Trig point

★ Trilateration, where a point is calculated given its distances from other known points