Calculate area, perimeter, all angles and side lengths for any triangle. Enter any known values to solve the rest.
Enter any two values — the third will be calculated.
A triangle is fully defined when three independent measurements are known — any combination of sides and angles (provided at least one side is given). The triangle calculator solves for all remaining properties: sides, angles, area, perimeter and triangle type. It supports four input methods corresponding to the standard geometric cases: SSS (three sides), SAS (two sides and the included angle), ASA (two angles and the side between them), and the special case of right triangles using the Pythagorean theorem. Every triangle property can be calculated from these inputs using the laws of sines, cosines and Heron's formula.
Two fundamental theorems make all triangle calculations possible. The law of sines states that the ratio of any side to the sine of its opposite angle is constant: a/sin(A) = b/sin(B) = c/sin(C). This allows finding unknown sides and angles in ASA and AAS configurations. The law of cosines is a generalisation of the Pythagorean theorem: c^2 = a^2 + b^2 - 2ab x cos(C). When C = 90 degrees, cos(90) = 0 and the equation reduces to c^2 = a^2 + b^2, the Pythagorean theorem. The law of cosines handles SSS (finding angles) and SAS (finding the third side). Together these two laws solve any non-degenerate triangle. Use our square root calculator when working through the Pythagorean theorem manually, and our area calculator for other shape area calculations beyond triangles.
| Known Inputs | Case | Method | Solves For |
|---|---|---|---|
| 3 sides (a, b, c) | SSS | Law of cosines + Heron's formula | All angles, area, perimeter |
| 2 sides + included angle | SAS | Law of cosines then sines | Third side, other angles, area |
| 2 angles + any side | ASA / AAS | Law of sines | Third angle, other sides, area |
| Right triangle, 2 sides | Right | Pythagorean theorem | Third side, angles, area |
| 2 sides + non-included angle | SSA | Law of sines (ambiguous) | May have 0, 1, or 2 solutions |
Heron's formula calculates triangle area from all three side lengths without requiring the height, making it extremely useful when the height is not directly measurable. The formula: first calculate the semi-perimeter s = (a + b + c) / 2, then Area = sqrt(s x (s-a) x (s-b) x (s-c)). Example: triangle with sides 5, 7 and 8. Semi-perimeter s = (5+7+8)/2 = 10. Area = sqrt(10 x 5 x 3 x 2) = sqrt(300) = 17.32 square units. Heron's formula is attributed to Heron of Alexandria (around 60 AD) though it may have been known earlier. It is particularly useful in surveying and construction where all side lengths can be measured directly but heights cannot. Use our square root calculator to compute the square root step in Heron's formula manually, and our percentage calculator to compare triangle areas as percentages.
Triangles are classified in two independent systems. By side lengths: equilateral triangles have all three sides equal and all three angles equal to 60 degrees — they are a special case of both isosceles and acute triangles. Isosceles triangles have exactly two equal sides, with the two angles opposite those sides also equal. Scalene triangles have all three sides and all three angles different. By interior angles: acute triangles have all three angles less than 90 degrees. Right triangles have exactly one 90-degree angle, making the law of cosines reduce to the Pythagorean theorem and enabling trigonometric ratios (sin, cos, tan) to be defined directly. Obtuse triangles have exactly one angle greater than 90 degrees. A triangle can belong to one category from each system simultaneously — an isosceles right triangle (like the 45-45-90 triangle used extensively in geometry) or a scalene acute triangle are both common. The sum of interior angles is always exactly 180 degrees for any flat (Euclidean) triangle regardless of type or size.
| Triangle Type | Side Condition | Angle Condition | Area Formula Available |
|---|---|---|---|
| Equilateral | a = b = c | A = B = C = 60 deg | A = (sqrt(3)/4) x s^2 |
| Isosceles | a = b, c different | A = B, C different | Heron or 0.5 x b x h |
| Scalene | All different | All different | Heron or SAS formula |
| Right | a^2 + b^2 = c^2 | One angle = 90 deg | A = 0.5 x leg1 x leg2 |
| Obtuse | Any combo | One angle > 90 deg | Heron or SAS formula |
| Acute | Any combo | All angles < 90 deg | Heron or SAS formula |
The Pythagorean theorem (a^2 + b^2 = c^2) is the most famous theorem in mathematics and the foundation of Euclidean geometry. Several special right triangles with integer or simple ratio side lengths appear repeatedly in geometry, trigonometry and practical measurement. The 3-4-5 triangle: 3^2 + 4^2 = 9 + 16 = 25 = 5^2. This is the most commonly used Pythagorean triple. Carpenters use the 3-4-5 ratio to check whether corners are square — if a corner measuring 3 feet by 4 feet has a diagonal of exactly 5 feet, the corner is a right angle. The 5-12-13 triple: 5^2 + 12^2 = 25 + 144 = 169 = 13^2. The 8-15-17 triple. The 45-45-90 triangle: legs are equal and the hypotenuse is leg x sqrt(2). The 30-60-90 triangle: sides are in ratio 1 : sqrt(3) : 2, originating from the equilateral triangle bisected. Our fraction calculator helps when working with ratio-based triangle dimensions, and our ratio calculator handles proportional scaling of any triangle.
Triangle geometry underlies an enormous range of real-world applications. Construction and surveying: the 3-4-5 method ensures square corners, and triangulation allows distances to inaccessible points to be measured using only angles and one known baseline. Navigation: GPS triangulation uses signal time differences from multiple satellites (effectively the SSS or SAS case in three dimensions) to determine position to within metres. Architecture and engineering: roof pitch calculations, truss design and structural load analysis all involve triangle geometry. The most stable structural shape is a triangle — a rigid polygon that cannot be deformed without changing side lengths. In contrast, squares and rectangles can be pushed into parallelograms without changing side lengths. This is why triangular trusses are used in bridges, roofs and towers. Computer graphics: all 3D objects are rendered as collections of triangles (triangle meshes) because any polygon can be decomposed into triangles and GPUs are optimised to process triangles efficiently. Use our area calculator for related shape calculations and our percentage calculator to compare measurements as proportions.
The triangle inequality theorem — which states that the sum of any two sides of a triangle must be greater than the third side — is the fundamental constraint that determines whether three given lengths can form a valid triangle. If a=3, b=4, c=9, then 3+4=7 which is less than 9, so no triangle exists with these dimensions. Our calculator checks this condition automatically and alerts you if the inputs cannot form a valid triangle. This constraint has a beautiful geometric interpretation: if you tried to physically join three rods of these lengths, the two shorter ones together would not reach from one end of the longer rod to the other. Use our square root calculator for Pythagorean theorem calculations and our fraction calculator when working with fractional side lengths or angle measurements. Triangles appear throughout nature and human design in ways that reflect their unique geometric properties. In nature, the triangular arrangement of atoms in crystalline structures provides the most efficient close-packing. Spiderwebs use triangular sections for maximum strength-to-material-weight ratio. Mountain peaks form triangular profiles in cross-section because erosion removes material in ways that naturally produce stable triangular slopes. In human design, the Eiffel Tower uses triangular lattice structures throughout because triangles distribute load more efficiently than any other polygon. Road signs are triangular in many countries specifically because the shape is immediately distinctive from circular or rectangular signs. The ancient Egyptians used the 3-4-5 right triangle to lay out the square bases of the pyramids — measuring ropes with knots at intervals of 3, 4 and 5 units to construct perfect right angles without modern tools. The same technique is still used by construction workers today to verify corners are square on large-scale projects. Our area calculator covers all other common shapes alongside triangles, and our ratio calculator handles proportional scaling of triangle dimensions for any project that requires maintaining triangle proportions at a different size.