Which formula do you use to solve a triangle based on what information you have?

The approach depends entirely on what's given: SSS (three sides known) — use the Law of Cosines to find each angle: cos(A) = (b² + c² - a²)/(2bc). Calculate the largest angle first (opposite the longest side) to check for obtuse triangles. SAS (two sides and the included angle) — use the Law of Cosines to find the third side: c² = a² + b² - 2ab×cos(C). Then use the Law of Sines to find another angle. ASA (two angles and the included side) — the third angle = 180° - A - B. Then use the Law of Sines: a/sin(A) = b/sin(B) = c/sin(C) to find the other sides. AAS (two angles and a non-included side) — same approach as ASA since the third angle is determined. SSA (two sides and a non-included angle) — the ambiguous case. Use the Law of Sines to find the angle opposite the known side: sin(B) = b×sin(A)/a. This may yield zero, one, or two valid triangles. If the calculated sin(B) > 1: no triangle exists. If sin(B) = 1: one right triangle. If sin(B) < 1: check both B and (180° - B) — if both give a valid triangle (angles sum to less than 180°), two solutions exist. AAA (three angles only) — determines the shape but not the size. Infinite similar triangles exist; a side length is needed.

How does the Law of Cosines relate to the Pythagorean theorem?

The Law of Cosines is a generalization of the Pythagorean theorem that works for all triangles, not just right triangles: c² = a² + b² - 2ab×cos(C). When angle C = 90°, cos(90°) = 0, and the formula reduces to c² = a² + b² — exactly the Pythagorean theorem. When C 90° (obtuse angle), cos(C) is negative, so -2ab×cos(C) becomes positive, making c² larger than a² + b². The side opposite an obtuse angle is longer. Practical accuracy tip: when using the Law of Cosines to find angles, calculate the angle opposite the longest side first. If this angle is obtuse, you know immediately; the other two angles must be acute. When using the Law of Sines afterward, arcsine only returns values in [0°, 90°], so it can miss obtuse angles — but since a triangle has at most one obtuse angle, and you've already found it, the remaining Law-of-Sines calculations are unambiguous. Numerical precision: for very flat or nearly degenerate triangles (one angle close to 0° or 180°), the Law of Cosines can lose precision. Use the alternative form with the haversine for better numerical stability in such edge cases.

What is the ambiguous case (SSA) in triangle solving?

The ambiguous case occurs when you are given two sides and a non-included angle (SSA). Depending on the specific lengths of the sides and the angle, there can be zero, one, or two possible triangles that satisfy the given conditions. For instance, if side 'a' is opposite angle 'A', and side 'b' is given, calculating `h = b * sin(A)` helps determine the number of solutions: if `a = b`, one triangle exists; and if `h < a < b`, two distinct triangles exist.

When is the Law of Sines used, and what is its formula?

The Law of Sines is primarily applied to solve triangles when you know two angles and one side (AAS or ASA cases), or in some instances of the ambiguous SSA case. It states that the ratio of the length of a side to the sine of its opposite angle is constant for all three sides: `a / sin(A) = b / sin(B) = c / sin(C)`. For example, if angles A=40°, B=60°, and side a=10 are known, side b can be found using `b = a * sin(B) / sin(A) = 10 * sin(60°) / sin(40°)`, which yields approximately 13.47.

How do you determine if three given side lengths can form a valid triangle?

For three side lengths to form a valid triangle, they must satisfy the Triangle Inequality Theorem, which states that the sum of the lengths of any two sides must be greater than the length of the third side. For example, given sides 7, 10, and 12, it forms a valid triangle because `7+10 > 12`, `7+12 > 10`, and `10+12 > 7`. However, sides 3, 4, and 8 cannot form a triangle because `3+4` (which is 7) is not greater than `8`.

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Triangle Solver

Enter known values (SSS, AAS, ASA). Angles in degrees.

Side a

Side b

Side c

Angle A (°)

Angle B (°)

Angle C (°)

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متغیر کی تشریح

a,b,c= side lengthsA,B,C= angles opposite sides a, b, cs= semi-perimeter = (a+b+c)/2

Law of sines

Relates sides and their opposite angles.

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Law of cosines

Generalisation of Pythagorean theorem.

Find side c

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Find angle C

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Area (Heron's formula)

Area from three sides only.

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Area

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Area from base and height

Standard formula.

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Deep Dive

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