Triangle Calculator

What Is a Triangle Calculator?

A triangle calculator determines the missing sides, angles, area, and perimeter of any triangle given sufficient known values. It applies the law of sines, law of cosines, and Heron's formula to solve oblique and right triangles.

Triangle geometry is foundational to mathematics—Euclid's Elements (c. 300 BCE) devotes two of thirteen books to triangles¹. The interior angles of any triangle always sum to exactly 180°, a property proven in Euclidean geometry². Heron's formula, dating to 60 CE, calculates area from three side lengths alone³. The Law of Cosines generalizes the Pythagorean theorem to all triangle types⁴.

Triangle Formulas and Properties

Area of any triangle = ½ × base × height. When three sides are known, use Heron's formula: Area = √(s(s−a)(s−b)(s−c)) where s = (a+b+c)/2. For SAS (two sides and included angle): Area = ½ × a × b × sin(C). The Pythagorean theorem (a² + b² = c²) applies only to right triangles. For non-right triangles, use the Law of Cosines: c² = a² + b² − 2ab·cos(C), or the Law of Sines: a/sin(A) = b/sin(B) = c/sin(C).

Triangle Types and the Angle Sum Rule

Interior angles always sum to 180°. Right: one 90° angle. Acute: all angles below 90°. Obtuse: one angle above 90°. Equilateral: all sides equal, all angles 60°. Isosceles: two equal sides and two equal base angles. The longest side is always opposite the largest angle. The Triangle Inequality Theorem states any two sides must sum to more than the third — violating this means no valid triangle can be formed with those dimensions.

Triangle Fundamentals and Classification

Triangles are the most fundamental polygon — the simplest closed shape with straight sides and the basis for structural engineering, trigonometry, and computational geometry. Every triangle has three sides, three angles that sum to exactly 180°, and a unique set of properties determined by these measurements. This calculator solves triangles using any sufficient combination of inputs: three sides (SSS), two sides and the included angle (SAS), two angles and a side (AAS or ASA), or side-angle-side combinations. The Law of Cosines (c² = a² + b² − 2ab·cos(C)) handles SSS and SAS cases, while the Law of Sines (a/sin(A) = b/sin(B) = c/sin(C)) resolves angle-related cases. Understanding which formula applies to each input combination eliminates the trial-and-error approach that slows manual triangle solving.

Triangle Types and Properties

Heron's Formula for Area

When you know all three side lengths but not the height, Heron's formula calculates area without requiring any angle or altitude measurement. First compute the semi-perimeter: s = (a + b + c)/2. Then area = √(s(s−a)(s−b)(s−c)). For a triangle with sides 7, 10, and 12: s = 14.5, area = √(14.5 × 7.5 × 4.5 × 2.5) = √2,440.3 ≈ 34.86 square units. Heron's formula is particularly valuable in surveying, where land parcels are measured by their boundary lengths rather than interior dimensions. It also provides a verification method: if the formula produces a negative value under the square root, the three lengths cannot form a valid triangle — a useful check for measurement errors in field work. The triangle inequality theorem states that the sum of any two sides must exceed the third side, and Heron's formula implicitly enforces this constraint.

Right Triangles and the Pythagorean Theorem

Right triangles — those containing exactly one 90° angle — are the foundation of trigonometry and appear in virtually every practical measurement scenario. The Pythagorean theorem (a² + b² = c², where c is the hypotenuse) relates the three sides and enables calculation of any unknown side from the other two. Common right triangle ratios appear so frequently in construction and engineering that they are memorized: 3-4-5 (and its multiples 6-8-10, 9-12-15), 5-12-13, and 8-15-17. Carpenters use the 3-4-5 ratio to verify right angles: measuring 3 feet along one wall and 4 feet along the adjacent wall, the diagonal should measure exactly 5 feet if the corner is square. Trigonometric functions — sine, cosine, and tangent — express the relationships between right triangle sides and angles: sin(θ) = opposite/hypotenuse, cos(θ) = adjacent/hypotenuse, tan(θ) = opposite/adjacent. These ratios extend triangle calculations to any angle, enabling measurement of heights, distances, and angles that cannot be directly measured. Use our Area Calculator for quick area computations and our Perimeter Calculator for perimeter-related problems.

Triangles in Structural Engineering

The triangle is the only polygon that is inherently rigid — a triangle made of rigid bars with pinned joints cannot deform without breaking a bar, while a square or rectangle can collapse into a parallelogram. This structural rigidity makes triangular truss systems the foundation of bridges, roof structures, transmission towers, and crane booms. The forces in each truss member (tension or compression) can be calculated by decomposing external loads into triangular force diagrams at each joint. Triangulation — the division of complex shapes into networks of triangles — is also the basis of geodetic surveying, 3D computer graphics (all surfaces are rendered as triangle meshes), and finite element analysis (engineering simulation software divides complex structures into triangular or tetrahedral elements for stress analysis). Understanding triangle geometry is therefore not merely academic — it underpins the structural integrity of nearly every engineered structure and the visual rendering of every 3D digital environment.

Special Triangles and Memorized Ratios

Two special triangle types appear so frequently in mathematics and engineering that their ratios are memorized. The 45-45-90 triangle (isosceles right triangle) has side ratios of 1:1:√2 — if the legs are each 1 unit, the hypotenuse is √2 ≈ 1.414 units. The 30-60-90 triangle has side ratios of 1:√3:2 — the side opposite 30° is half the hypotenuse, and the side opposite 60° is (√3/2) times the hypotenuse. These ratios enable instant calculation without trigonometric functions: a 30-60-90 triangle with hypotenuse 10 has legs of exactly 5 and 5√3 ≈ 8.66. In construction, a 30-60-90 triangle creates the slope of a standard staircase, and 45-45-90 triangles define diagonal bracing angles. Equilateral triangles (all angles 60°) have the special property that height = (√3/2) × side, enabling quick area calculation: area = (√3/4) × s² for side length s. A 6-inch equilateral triangle has area (√3/4) × 36 ≈ 15.59 square inches.

Triangle Congruence and Similarity

Two triangles are congruent (identical in shape and size) when any of these conditions holds: SSS (all three sides equal), SAS (two sides and the included angle equal), ASA (two angles and the included side equal), or AAS (two angles and any corresponding side equal). Congruence is fundamental in construction — verifying that structural members are identical ensures consistent load-bearing capacity. Similar triangles (same shape, different size) have proportional sides and equal angles. Similarity enables indirect measurement: by measuring a short shadow and comparing it to a building's shadow at the same time of day, the ratio of shadow lengths equals the ratio of heights — a technique used since ancient Egypt to measure the pyramids. In engineering, scale models exploit similarity: a 1:100 scale bridge model with a 2-meter span represents a 200-meter actual bridge, and forces scale proportionally. See our Ratio Calculator for proportion computations used in similarity problems.

Solving Oblique Triangles

Triangles without a right angle (oblique triangles) require the Law of Sines or Law of Cosines rather than the Pythagorean theorem. The Law of Cosines generalizes the Pythagorean theorem to all triangles and is used when you know three sides (SSS) or two sides and their included angle (SAS). The Law of Sines is used when you know two angles and any side (AAS/ASA) or two sides and an angle opposite one of them (SSA — the ambiguous case, which can produce zero, one, or two valid triangles). Recognizing the ambiguous case prevents errors: when given two sides and a non-included angle, always check whether the given information permits multiple solutions before reporting a single answer. In navigation, oblique triangle solutions enable position fixing from bearing measurements to known landmarks — a technique used in marine and aerial navigation long before GPS and still taught as a backup method.

How to Use This Calculator

1. Enter known measurements — Input any combination of sides and angles — at least three values (with at least one side) are needed to solve a triangle completely.
2. Select the appropriate case — The calculator detects whether you've provided SSS (three sides), SAS (two sides + included angle), ASA, AAS, or SSA — and applies the correct solving method.
3. Review all solved values — The calculator shows all three sides, all three angles, area, perimeter, height, and the circumradius/inradius. For SSA (ambiguous case), it shows both possible solutions.
4. Verify with angle sum — All interior angles must sum to exactly 180°. The calculator uses this as an automatic validation check and warns if your inputs are geometrically impossible.

Tips and Best Practices

See also: Pythagorean Theorem Calculator · Area Calculator · Circle Calculator