Triangle Calculator

Triangle types: equilateral, isosceles, scalene

• Equilateral: all three sides equal. All three angles equal 60°.
• Isosceles: exactly two sides equal. The two angles opposite those sides are also equal.
• Scalene: all three sides different. All three angles different.

By angle: acute, right, obtuse

• Acute: all three angles less than 90°.
• Right: one angle equals exactly 90°. The side opposite the right angle is the “hypotenuse.”
• Obtuse: one angle greater than 90° (and less than 180°).

A triangle's three angles always sum to 180°. The two classifications combine: a triangle is “equilateral acute” or “isosceles right” etc.

Pythagorean theorem

The most-known geometry result: in any right triangle, c² = a² + b², where c is the hypotenuse. Used since at least 1800 BCE in Babylonian texts; named after the 6th-century BCE Greek philosopher Pythagoras.

The 3-4-5 triangle is the smallest integer right triangle. Other Pythagorean triples: 5-12-13, 8-15-17, 7-24-25, 20-21-29. There are infinitely many. Useful in carpentry — to make sure a corner is square, measure 3 ft along one wall, 4 ft along the perpendicular wall; the diagonal between them should be exactly 5 ft.

Law of Cosines

Generalizes Pythagorean to any triangle: c² = a² + b² − 2ab·cos(C). When C = 90°, cos(C) = 0, and you get back to Pythagorean. For obtuse angles (C > 90°), cos(C) is negative, making c² larger. For acute angles (C < 90°), cos(C) is positive, making c² smaller.

Used in SAS mode (you know two sides and the included angle, want the third side) and SSS mode (rearranged to find an angle from three sides).

Law of Sines

a/sin(A) = b/sin(B) = c/sin(C) = 2R, where R is the circumradius (radius of the circle through all three vertices). Useful when you know an angle-side opposite pair plus another piece (ASA, AAS, SSA).

SSA is the “ambiguous case” — sometimes there are two valid triangles for the same inputs. The calculator focuses on Pythagorean / SSS / SAS, which all have unique solutions.

Heron's formula

Area from three sides only: A = √(s(s-a)(s-b)(s-c)) where s = (a+b+c)/2. Heron of Alexandria, ~60 CE. Beautiful because it doesn't need any angle or height — just the sides.

For 3-4-5: s = 6, area = √(6 × 3 × 2 × 1) = √36 = 6 (verify: half × base × height = 0.5 × 3 × 4 = 6 ✓).

When triangle calculations are useful

• Carpentry and construction: roof rafters, stair stringers, square corners (3-4-5 method), gable framing.
• Land surveying: triangulation to find distances and elevations.
• Navigation: triangulation from known reference points.
• Computer graphics: every 3D model is built from triangles.
• Engineering: trusses, bracing, anything that resists deformation.
• Astronomy: parallax measurement, stellar distance.

For other geometry tools: Square Footage Calculator for areas of common shapes, Unit Converter for measurement conversions.