The triangle is the simplest polygon and the most solved shape in mathematics: three sides and three angles, six quantities, of which any three (with at least one side) determine the rest. The methods for finding the rest are few and old — Pythagoras for right triangles, the sine and cosine rules for the rest, Heron’s formula for area from three sides — and they are the same methods used in surveying, navigation, construction, and computer graphics, where triangles are the atoms everything else is built from. This guide states each method, works it, and names its condition: which method applies depends on what is known, and using the wrong one is the commonest error in triangle problems.

Types: by sides, by angles, and what each type promises

By sides: equilateral — all three sides equal, all three angles 60°. Isosceles — two sides equal, with the angles opposite them equal. Scalene — all sides different, all angles different. By angles: acute — all angles under 90°. Right — one angle exactly 90°, with the side opposite it, the hypotenuse, the longest side. Obtuse — one angle over 90°. The classifications combine — a triangle can be right and isosceles, or obtuse and scalene, but never right and obtuse, and never equilateral and right, because the angle sum forbids it.

The classification is not taxonomy for its own sake: it states which methods apply. A right triangle admits Pythagoras and the basic trigonometric ratios. An isosceles triangle’s symmetry halves the problem — the altitude from the apex bisects the base and the apex angle. An equilateral triangle’s area follows from one side alone. Knowing the type before choosing the method is the step that prevents applying Pythagoras to a triangle with no right angle in it.

Angles: the 180° rule and the exterior angle

The interior angles sum to 180° in Euclidean geometry — the geometry of flat surfaces, which is the geometry of every practical triangle problem. Two known angles give the third by subtraction: angles of 50° and 60° leave 70°. The exterior angle at any vertex equals the sum of the two interior angles opposite it, which is the same rule restated and the quicker route in problems that give an exterior angle. The angle sum is also the check on every triangle solution: a solved triangle whose angles do not sum to 180° is wrong somewhere, and the check should be run before the answer is trusted.

Pythagoras and the right triangle

In a right triangle, a² + b² = c², where c is the hypotenuse. With legs 3 and 4: c² = 9 + 16 = 25, so c = 5. With hypotenuse 13 and one leg 5: the other leg is √(169 − 25) = √144 = 12. The theorem runs in both directions — it finds a missing side, and it tests whether a triangle is right: sides 3, 4, 5 satisfy the equation and the triangle is right; sides 2, 3, 4 do not (4 + 9 ≠ 16) and the triangle is not right. Pythagorean triples — 3-4-5, 5-12-13, 8-15-17, 7-24-25, and their multiples — are the integer cases, and recognising them saves a calculation.

The trigonometric ratios are Pythagoras’ companions: for an acute angle in a right triangle, sin = opposite/hypotenuse, cos = adjacent/hypotenuse, tan = opposite/adjacent. In the 3-4-5 triangle, the angle opposite the side of 3 has sin = 3/5 = 0.6, giving 36.87°; the other acute angle is 53.13°; and 36.87 + 53.13 + 90 = 180, as the angle sum requires. The trigonometry calculation on this site carries the ratio arithmetic, and the triangle calculation carries the full solution.

The sine rule and the cosine rule: every other triangle

The sine rule states that each side divided by the sine of its opposite angle is the same for all three sides: a/sin A = b/sin B = c/sin C. Given two angles and a side (AAS or ASA), it finds the rest directly: with A = 50°, B = 60° (so C = 70°), and a = 10, side b = 10 × sin 60° / sin 50° = 10 × 0.8660 / 0.7660 = 11.31, and c = 10 × sin 70° / sin 50° = 12.27. Given two sides and a non-included angle (SSA), the sine rule can give two answers — the ambiguous case: sin B computed from the data may correspond to two angles, B and 180° − B, both geometrically possible, and the problem must be checked for both, with the triangle’s other data deciding which survives.

The cosine rule states c² = a² + b² − 2ab·cos C, and it solves the two cases the sine rule handles badly: two sides and the included angle (SAS), and three sides (SSS). With a = 7, b = 9, and included angle C = 60°: c² = 49 + 81 − 2 × 7 × 9 × 0.5 = 130 − 63 = 67, so c = 8.19. With three sides, the rule rearranged gives the angles: cos C = (a² + b² − c²) / (2ab). The cosine rule is Pythagoras generalised — when C is 90°, cos C is 0, the correction term vanishes, and the equation is Pythagoras again.

Area: three formulas, one answer

Area = ½ × base × height, where the height is the perpendicular distance from the base to the opposite vertex. For the 3-4-5 right triangle with legs as base and height: ½ × 3 × 4 = 6. Heron’s formula gives the same answer from three sides alone: s = (a + b + c) / 2 = 6, area = √(s(s−a)(s−b)(s−c)) = √(6 × 3 × 2 × 1) = √36 = 6. The two methods agreeing is the check. A third form uses two sides and the included angle: area = ½ab·sin C, which for the 3-4-5 triangle with the included right angle gives ½ × 3 × 4 × 1 = 6 again. Three formulas, one answer — and a solution whose formulas disagree has an error in it, not a difference of opinion.

Frequently asked questions

How do you find the third angle of a triangle?

Subtract the two known angles from 180°. The interior angles of any triangle sum to 180° in Euclidean geometry, so two angles always determine the third. Angles of 50° and 60° leave 70°. The exterior angle at a vertex equals the sum of the two opposite interior angles.

How do you find the hypotenuse?

With Pythagoras: c = √(a² + b²), where a and b are the legs. Legs 3 and 4 give c = √25 = 5. Pythagoras applies only to right triangles — for any other triangle, use the cosine rule with the included angle, or the sine rule with the angles known.

What is the area of a 3-4-5 triangle?

Six square units: ½ × 3 × 4 = 6, taking the legs as base and height. Heron’s formula agrees: s = 6, area = √(6 × 3 × 2 × 1) = √36 = 6. The two methods agreeing is the check that the triangle and the arithmetic are both right.

When do you use the sine rule and when the cosine rule?

Use the sine rule when two angles and a side are known (AAS/ASA), or two sides and a non-included angle (SSA — watch for the ambiguous case, two possible triangles). Use the cosine rule for two sides and the included angle (SAS) or three sides (SSS). The cosine rule with a 90° angle reduces to Pythagoras.

What is the ambiguous case?

When the sine rule is given two sides and a non-included angle, the computed sine can correspond to two angles — B and 180° − B — and both may form valid triangles with the given data. Both must be checked against the other conditions of the problem; assuming the first answer is the only answer is the error.

Can three angles determine a triangle?

They determine the shape but not the size. Three angles fix the ratios of the sides, and any scaled copy has the same angles — the triangle is determined up to similarity. At least one side must be known to fix the size; three known quantities including one side determine the triangle completely.