Triangle Calculator
Solve any triangle from any three known values — sides, angles or a mix. Calculates area, perimeter, all angles and all sides with a step-by-step solution and a live visual diagram.
Triangle Calculator
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Step-by-step solution, visual diagram and SSA ambiguous case — what most solvers skip
Most triangle calculators return numbers without explaining how they were reached. This tool shows every formula applied, draws a scaled diagram of the solved triangle, and correctly identifies and handles the SSA ambiguous case where two valid triangles may exist.
How to solve a triangle
LazyTools vs other triangle calculators
Most free triangle calculators return numbers without explaining how they were reached. Step-by-step working, SSA ambiguous case handling and a live diagram together are absent from the major free options.
| Feature | ⭐ LazyTools | calculator.net | mathsisfun.com | omnicalculator.com |
|---|---|---|---|---|
| SSS, SAS, ASA, AAS modes | ✔ | ✔ | ✔ | ✔ |
| SSA ambiguous case | ✔ Both solutions | ⚠ Partial | ✘ | ✔ |
| Step-by-step solution | ✔ | ✘ | ⚠ Basic | ✔ |
| Live visual diagram | ✔ Scaled SVG | ⚠ Static | ✔ | ✘ |
| Triangle type classification | ✔ | ✔ | ✔ | ✔ |
| Right triangle mode | ✔ | ✔ | ✔ | ✔ |
| No ads / no signup | ✔ | ⚠ Ads | ⚠ Ads | ⚠ Ads |
Triangle formulas and laws
| Formula / Law | Expression | Used for |
|---|---|---|
| Law of Cosines | a² = b² + c² - 2bc cos(A) | SSS and SAS — finding angles from sides or missing side |
| Law of Sines | a/sin(A) = b/sin(B) = c/sin(C) | ASA, AAS, SSA — finding sides from angles |
| Angle sum | A + B + C = 180° | Finding third angle when two are known |
| Heron's formula | Area = sqrt(s(s-a)(s-b)(s-c)) | Area from three sides; s = (a+b+c)/2 |
| SAS area | Area = (1/2) ab sin(C) | Area from two sides and included angle |
| Pythagoras | a² + b² = c² | Right triangles only; c is hypotenuse |
| Perimeter | P = a + b + c | Sum of all three sides |
Triangle type classifications
| Type | Condition | Properties |
|---|---|---|
| Equilateral | a = b = c | All angles 60°; 3 lines of symmetry |
| Isosceles | Two sides equal | Two base angles equal; 1 line of symmetry |
| Scalene | All sides different | All angles different; no lines of symmetry |
| Right | One angle = 90° | Pythagoras applies; hypotenuse is longest side |
| Acute | All angles < 90° | All altitudes fall inside the triangle |
| Obtuse | One angle > 90° | One altitude falls outside the triangle |
Triangle Calculator — Area, Perimeter, Angles and All Sides Explained
A triangle is uniquely determined (up to congruence) when exactly three independent values are known, provided they form a consistent and valid configuration. The six measurable quantities of a triangle are three sides (a, b, c) and three angles (A, B, C). Any three of these values — in the right combination — are enough to calculate all six. The process of finding all six values from a partial set is called “solving the triangle”.
Triangle calculator area perimeter angles online free
Calculating all properties of a triangle from a given set of inputs follows a well-defined algorithm based on which combination is provided. Three sides (SSS) uses the Law of Cosines to find each angle: cos(A) = (b² + c² - a²) / (2bc). Once all angles are known, area is found using Heron’s formula: Area = sqrt(s(s-a)(s-b)(s-c)) where s is the semi-perimeter (a+b+c)/2. Two sides and the included angle (SAS) also uses the Law of Cosines, then the Law of Sines for the remaining angles. Two angles and any side (ASA or AAS) uses the angle sum to find the third angle, then the Law of Sines to find the remaining sides.
Right triangle calculator with all sides and angles
A right triangle has one angle equal to 90°. Knowing any two of the three sides, or one side and one acute angle, is sufficient to solve it completely. The three sides relate through Pythagoras’ theorem: a² + b² = c² where c is the hypotenuse. The angles relate to sides through trigonometric ratios: sin(A) = opposite/hypotenuse = a/c, cos(A) = adjacent/hypotenuse = b/c, tan(A) = opposite/adjacent = a/b. The two acute angles always sum to 90°. This tool’s Right triangle mode applies these relationships automatically.
How to solve a triangle calculator SSS SAS
The choice of formula depends on which values are known. SSS (three sides known): apply the Law of Cosines three times to find each angle, then use Heron’s formula for area. SAS (two sides and included angle known): apply the Law of Cosines once to find the third side, then the Law of Sines for the remaining angles. The included angle must be between the two known sides. If the angle is not between the two sides, use SSA mode instead. ASA (two angles and included side): find the third angle using the angle sum, then apply the Law of Sines twice. AAS (two angles and non-included side): same approach as ASA after finding the third angle.
Triangle law of cosines calculator
The Law of Cosines is a generalisation of Pythagoras’ theorem that applies to all triangles: c² = a² + b² - 2ab cos(C). When angle C is 90°, cos(90°) = 0 and the formula reduces to c² = a² + b². The Law of Cosines is used whenever two sides and the angle between them (SAS) are known — to find the third side — or when all three sides (SSS) are known — to find the angles. It is particularly useful because it avoids the SSA ambiguity that afflicts the Law of Sines in certain configurations.
Oblique triangle solver
An oblique triangle is any triangle that does not contain a right angle — it may be acute (all angles less than 90°) or obtuse (one angle greater than 90°). Solving oblique triangles always requires either the Law of Cosines or the Law of Sines, as Pythagoras’ theorem does not apply. The ambiguous case (SSA) is unique to oblique triangles: given two sides and an angle not between them, zero, one or two valid triangles may exist depending on whether the given side is shorter, equal to or longer than the height of the triangle from the opposite vertex.
Heron formula triangle area calculator
Heron’s formula calculates the area of a triangle from its three side lengths alone, without needing any angles: Area = sqrt(s(s-a)(s-b)(s-c)) where s is the semi-perimeter s = (a+b+c)/2. This is especially useful in SSS problems where angles are not initially known. For example, a triangle with sides 3, 4, 5 has s = 6, and Area = sqrt(6 x 3 x 2 x 1) = sqrt(36) = 6. An equivalent formula using two sides and the included angle is Area = (1/2)ab sin(C), which is used in SAS configurations and typically more numerically stable.