Triangle Calculator
Solve any triangle using SSS, SAS, ASA, AAS, or Right Triangle — get all sides, angles, area, and more.
SSS — Enter Known Values
Results
Side a
3
Side b
4
Side c
5
Angle A
36.87°
Angle B
53.13°
Angle C
90°
Area
6
Perimeter
12
Height hₐ
4
Height h_b
3
Height h_c
2.4
Triangle Calculator
Solve any triangle with sides and angles using the Law of Sines, Law of Cosines, and Heron's formula.
Guide
How it works
Foundations: The Pythagorean Theorem
The most famous relationship in geometry, a² + b² = c², applies to right triangles where c is the hypotenuse. Pythagorean triples — integer solutions like (3, 4, 5), (5, 12, 13), (8, 15, 17) — were known to Babylonians 1,000 years before Pythagoras. The theorem extends to all triangles through the Law of Cosines: c² = a² + b² − 2ab·cos(C), which reduces to the Pythagorean theorem when C = 90° since cos(90°) = 0.
SOHCAHTOA and Trigonometry
In a right triangle, sin(θ) = Opposite/Hypotenuse, cos(θ) = Adjacent/Hypotenuse, tan(θ) = Opposite/Adjacent. This mnemonic (SOHCAHTOA) is the gateway to trigonometry. It enables surveyors to calculate inaccessible distances, architects to compute roof pitches, and navigators to plot bearings.
Law of Sines and Law of Cosines
The Law of Sines (a/sin A = b/sin B = c/sin C) allows solving triangles when you know two angles and a side (AAS, ASA) or two sides and a non-included angle (SSA — the ambiguous case). The Law of Cosines handles SSS and SAS cases. Both laws are generalizations of right-triangle trigonometry to any triangle.
Heron's Formula
Greek mathematician Heron of Alexandria (~60 AD) derived the area formula: Area = √(s(s−a)(s−b)(s−c)) where s = (a+b+c)/2 is the semi-perimeter. This remarkable formula computes area knowing only the three sides — no angles needed. It's used in computer graphics for triangle mesh area calculations and in GPS positioning algorithms.
Triangle Inequality and Classification
For a valid triangle, any side must be less than the sum of the other two (triangle inequality). Triangles are classified by sides: equilateral (all sides equal, all angles 60°), isosceles (two sides equal), scalene (all sides different). By angles: acute (all <90°), right (one =90°), obtuse (one >90°). The largest angle always opposes the longest side.
Engineering Applications
Structural trusses — bridges and roofs rely on triangles as the only rigid polygon (a rectangle deforms under load; a triangle doesn't). GPS triangulation determines position by measuring distances from three satellites, then solving the resulting triangle system. Navigation: Pilots use the wind triangle to compute ground speed and heading when wind pushes an aircraft off course. Surveying: Triangulation networks cover entire continents, with each triangle's angles summing to 180°.
The Ambiguous Case (SSA)
When given two sides and a non-included angle, there may be 0, 1, or 2 valid triangles — called the ambiguous case. If the side opposite the given angle is shorter than the other given side, two triangles may exist (two possible positions of the third vertex). This is why SSA is not a valid congruence criterion (unlike SSS, SAS, ASA, AAS).
What is the triangle inequality theorem?expand_more
For any triangle with sides a, b, c, the sum of any two sides must be greater than the third side: a + b > c, a + c > b, and b + c > a. If this condition is violated, no triangle can be formed. For example, sides 1, 2, 10 cannot form a triangle because 1 + 2 = 3 < 10.
How do I find the area of a triangle without the height?expand_more
Use Heron's formula: compute s = (a + b + c) / 2, then Area = √(s(s−a)(s−b)(s−c)). Alternatively, if you know two sides and the included angle: Area = 0.5 × a × b × sin(C). Both methods avoid needing the perpendicular height.
Why does the sum of angles in a triangle equal 180°?expand_more
This holds for triangles on a flat (Euclidean) plane. Draw a line parallel to one side through the opposite vertex — alternate interior angles show the three angles fit perfectly on a straight line (180°). On a sphere (non-Euclidean geometry), triangle angles sum to more than 180°.
What are Pythagorean triples?expand_more
Pythagorean triples are sets of three positive integers (a, b, c) satisfying a² + b² = c². Common examples: (3,4,5), (5,12,13), (8,15,17), (7,24,25). Any multiple of a triple also works (6,8,10 is 2×(3,4,5)). They're used in construction to create perfect right angles.
What is the difference between similar and congruent triangles?expand_more
Congruent triangles have exactly the same size and shape — all corresponding sides and angles are equal. Similar triangles have the same shape but different sizes — all angles are equal, and corresponding sides are proportional. Similarity is established by AA (Angle-Angle), SAS, or SSS similarity criteria.