Triangle Calculator

These abbreviations describe which triangle measurements you know: S = Side, A = Angle. SSS (three sides): use Law of Cosines to find angles. SAS (two sides + included angle): use Law of Cosines for the third side, then Law of Sines for remaining angles. ASA (two angles + included side): the third angle = 180° minus the other two, then Law of Sines. AAS (two angles + non-included side): same as ASA once you find the third angle. AAA (three angles) is the only case that doesn't uniquely determine a triangle — infinitely many similar triangles satisfy three angles.

The Law of Cosines states: c² = a² + b² − 2ab·cos(C). It generalizes the Pythagorean theorem: when C = 90°, cos(90°) = 0 and it reduces to c² = a² + b². Three equivalent forms: a² = b² + c² − 2bc·cos(A), b² = a² + c² − 2ac·cos(B), c² = a² + b² − 2ab·cos(C). To find an angle given three sides: cos(A) = (b² + c² − a²) / (2bc). The Law of Cosines works for any triangle, including obtuse triangles.

The Law of Sines states: a/sin(A) = b/sin(B) = c/sin(C). This common ratio equals the diameter of the triangle's circumscribed circle (circumradius × 2). Use it when you know: two angles and any side (ASA or AAS). Example: A = 30°, B = 70°, a = 5. C = 180° − 30° − 70° = 80°. b = a × sin(B)/sin(A) = 5 × sin(70°)/sin(30°) ≈ 9.40. Caution: when given two sides and a non-included angle (SSA), the Law of Sines can produce two solutions (the ambiguous case).

Several methods: (1) Base × height / 2: Area = ½ × b × h. (2) Two sides and included angle: Area = ½ × a × b × sin(C). (3) Heron's formula (three sides): s = (a+b+c)/2, Area = √(s(s−a)(s−b)(s−c)). (4) Coordinates: if vertices are (x₁,y₁), (x₂,y₂), (x₃,y₃), Area = ½|x₁(y₂−y₃) + x₂(y₃−y₁) + x₃(y₁−y₂)|. This calculator uses all three methods depending on which values are available.

An obtuse triangle has one angle greater than 90°. The Law of Cosines still applies, but the cosine of an obtuse angle is negative, so c² = a² + b² − 2ab·cos(C) gives a smaller value when the included angle is obtuse. When using the Law of Sines in the SSA case, check whether the second solution is valid: if A + the computed angle < 180°, a second valid triangle may exist. The Pythagorean theorem only applies to right triangles; for obtuse triangles, the longest side is always opposite the obtuse angle and c² > a² + b².

In Euclidean (flat) geometry, the interior angles of any triangle sum to exactly 180°. Proof: draw a line parallel to one side through the opposite vertex. The two alternate interior angles equal the base angles (by parallel line properties). The three angles together form a straight line = 180°. In non-Euclidean geometry (curved surfaces), this is not true: on a sphere, triangle angles sum to more than 180°; on a saddle surface (hyperbolic geometry), they sum to less. GPS triangulation must account for Earth's curved surface.

The triangle inequality states that the sum of any two sides must be greater than the third side: a+b > c, a+c > b, and b+c > a. If any condition fails, no valid triangle can be formed. Examples: sides 3, 4, 5: 3+4=7>5, 3+5=8>4, 4+5=9>3 — valid triangle. Sides 1, 2, 10: 1+2=3<10 — invalid. This theorem also applies to vectors and distances in more abstract mathematical settings (metric spaces).

Heron's formula calculates triangle area from three sides without needing height. Named after Hero of Alexandria (c. 60 AD): s = (a+b+c)/2 (semi-perimeter). Area = √(s(s−a)(s−b)(s−c)). Example: sides 3, 4, 5. s = (3+4+5)/2 = 6. Area = √(6×3×2×1) = √36 = 6 square units. Cross-check: right triangle with legs 3 and 4. Area = ½×3×4 = 6. Both methods agree. Heron's formula is especially useful in surveying and cartography when a triangle's vertices are known but no direct height measurement is available.

By sides: Equilateral (all sides equal, all angles 60°), Isosceles (two sides equal, two angles equal), Scalene (all sides different, all angles different). By angles: Acute (all angles < 90°), Right (one angle = 90°, satisfies a²+b²=c²), Obtuse (one angle > 90°). A triangle can be simultaneously classified by both: e.g., a right isosceles triangle has two equal legs and angles 45°-45°-90°. Special right triangles: 30°-60°-90° (sides in ratio 1:√3:2) and 45°-45°-90° (sides in ratio 1:1:√2).

A triangle has three heights (altitudes), one from each vertex perpendicular to the opposite side. Height on base a: hₐ = 2×Area/a. Height on base b: hᵇ = 2×Area/b. Height on base c: hᶜ = 2×Area/c. Alternatively using trigonometry: hₐ = b×sin(C) = c×sin(B). All three altitudes of any triangle meet at a single point called the orthocenter. For an acute triangle the orthocenter is inside; for a right triangle it is at the right-angle vertex; for an obtuse triangle it is outside the triangle.