Triangle Calculator
Last updated: 27 June 2026
Reviewed by Gavin Meiring, Lead research and primary author · Doctoral Candidate (Corporate Governance) · Research and drafting assisted by AI
- Pythagoras wasn't the first to discover his theorem. Babylonian clay tablets from 1000 years before Pythagoras show Pythagorean triples. He may have been the first to formally prove it.
- In non-Euclidean geometry (used in GPS and general relativity), the angles of a triangle don't add up to 180°. On a sphere, they can add up to 270° or more.
- The strongest shape in construction is the triangle. Triangular trusses are used in bridges, roofs, and the Eiffel Tower because triangles cannot deform without changing the length of a side.
Triangle Calculator
A triangle calculator finds all sides, angles, area, and perimeter of any triangle from a minimum of three known values. It is used by students, surveyors, engineers, and designers working with geometry, construction layouts, and navigation problems.
How to Use the Triangle Calculator
- Select what you know: three sides (SSS), two sides and the included angle (SAS), two angles and a side (AAS or ASA), or a right triangle with two known values.
- Enter your known values in the appropriate fields, using degrees for angles.
- Click Calculate to see all remaining sides, angles, area, and perimeter.
- Review the classified type of triangle (equilateral, isosceles, scalene, right, acute, or obtuse).
- Use the diagram to confirm the labelling matches your scenario.
The Formula
For any triangle with sides a, b, c and opposite angles A, B, C: angles sum to 180 degrees (A plus B plus C = 180). Law of sines: a divided by sin(A) = b divided by sin(B) = c divided by sin(C). Law of cosines: c squared = a squared plus b squared minus (2ab multiplied by cos(C)). Area using base and height: area = 0.5 multiplied by base multiplied by height. Area using Heron's formula: area = square root of (s(s minus a)(s minus b)(s minus c)), where s = (a plus b plus c) divided by 2. For right triangles: a squared plus b squared = c squared (Pythagorean theorem).
Real-World Example
A builder needs to cut a triangular roof section. Two sides measure 5.2 m and 3.8 m, and the included angle is 62 degrees. Using the law of cosines: c squared = 5.2 squared plus 3.8 squared minus (2 times 5.2 times 3.8 times cos(62)) = 27.04 plus 14.44 minus (39.52 times 0.4695) = 41.48 minus 18.55 = 22.93. c = square root of 22.93 = 4.79 m. Area = 0.5 multiplied by 5.2 multiplied by 3.8 multiplied by sin(62) = 0.5 times 19.76 times 0.8829 = 8.72 square metres.
Types of Triangles
Classified by sides: equilateral (all three sides equal, all angles 60 degrees); isosceles (two equal sides, two equal angles); scalene (all sides and angles different). Classified by angles: acute (all angles below 90 degrees); right (one angle exactly 90 degrees); obtuse (one angle above 90 degrees). No triangle can have two right angles or two obtuse angles because the angles must sum to exactly 180 degrees. The triangle inequality states that the sum of any two sides must be greater than the third side.
Three Cases Worked Through
| Case | Known values | Result |
|---|---|---|
| Three sides | a = 3, b = 4, c = 5 | right triangle, area 6, perimeter 12, angles 36.87, 53.13 and 90 degrees |
| Two sides and the included angle | a = 5.2, b = 3.8, C = 62 degrees | c = 4.7882, area 8.7235, perimeter 13.7882 |
| Two angles and a side | A = 40, B = 65, a = 6 | C = 75, b = 8.4598, c = 9.0163, area 24.5146 |
| Three sides, no right angle | a = 13, b = 14, c = 15 | area 84, angles 53.13, 59.49 and 67.38 degrees |
The last row is the reason Heron's formula earns its place. There is no height to measure and no right angle to lean on, and the area comes out exactly on a whole number.
Which Rule Applies
The calculator offers four starting points: three sides, two sides with the included angle, two angles with a side, or a right triangle with any two values. It does not offer two sides with a non-included angle, which is the one combination that can describe two different triangles. If that is all you know, measure the included angle or work both possibilities by hand and check each against a sketch.
Angles are entered in degrees and the three of them always sum to 180. The law of cosines returns the third side from two sides and the included angle. The law of sines does the rest, with one caution: when the known side is shorter than the side opposite the known angle, the inverse sine can return the supplementary angle as well, so treat a surprising result as a prompt to check rather than as an error.
Areas come from half the product of two sides and the sine of the included angle, or from Heron's formula when all three sides are known. The units follow what you entered, so a triangle measured in metres returns an area in square metres and a perimeter in metres.
Right Triangles That Need No Arithmetic
Some side combinations are whole numbers, and they turn up often enough in plans and workshops to be worth recognising without the calculator.
| Legs | Hypotenuse | Area | Perimeter |
|---|---|---|---|
| 3, 4 | 5 | 6 | 12 |
| 5, 12 | 13 | 30 | 30 |
| 8, 15 | 17 | 60 | 40 |
| 7, 24 | 25 | 84 | 56 |
| 20, 21 | 29 | 210 | 70 |
Each row satisfies a squared plus b squared equals c squared exactly, which is the test the calculator applies when it classifies a triangle as right-angled. A triangle that misses the test by a fraction of a unit, which is what a measurement usually does, comes back as acute or obtuse, so compare the angles as well as the label.
Frequently Asked Questions
What is the difference between the law of sines and the law of cosines? The law of sines is most useful when you know two angles and a side (AAS or ASA), or two sides and a non-included angle (SSA). The law of cosines works for any configuration but is most useful for SAS (two sides and the included angle) and SSS (all three sides). The law of cosines reduces to the Pythagorean theorem when one angle is 90 degrees.
Can a triangle have more than one obtuse angle? No. If one angle is greater than 90 degrees, the sum of the three angles would exceed 180 degrees if any other angle were also obtuse. A triangle can have at most one right angle and at most one obtuse angle.
What is the ambiguous case in triangle solving? The ambiguous case arises in SSA configurations (two sides and a non-included angle). Depending on the values, there may be zero, one, or two possible triangles. The calculator handles this by displaying all valid solutions and flagging when the ambiguous case applies.
How is triangle geometry used in real life? Triangles are the most structurally stable geometric shape, making them fundamental in bridge design, roof trusses, and geodesic domes. Surveyors use triangulation to measure land. Navigation and GPS rely on trilateration (a 3D extension of triangulation). Trigonometry derived from triangle geometry underlies signal processing, wave analysis, and computer graphics.
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Extended Reference Notes
The notes below cover the broader context that informs how to use the Triangle Calculator well.
Typical Input Ranges
Most real-world uses of the Triangle Calculator fall into a middle band where the result is stable and useful. Very small inputs to the Triangle Calculator often round to zero or near-zero, and very large inputs amplify every rounding error in the calculation. The middle band, where the Triangle Calculator inputs are ordinary sizes, is where the tool is most reliable.
Assumptions Behind the Formula
The Triangle Calculator assumes the inputs stay fixed across the period or scenario being modelled. Rates move, values change, and fees appear, so treat the Triangle Calculator output as a clean reference and layer in the frictions your own situation adds.
Common Edge Cases
Three situations change the Triangle Calculator answer in ways the formula does not surface: boundary values near zero, rounding cascades across many steps, and unit mismatches between fields. When any of these apply, sanity-check the Triangle Calculator result against an independent estimate.
When to Revisit the Calculation
The Triangle Calculator output is only as current as its inputs, so re-run the calculation whenever a key value changes materially. A quarterly re-check of the Triangle Calculator suits personal planning; monthly suits active business or investment decisions.
Relationship to Other Tools
The Triangle Calculator shares inputs and outputs with the other tools in its category. If the same numbers feed several tools, capture them once and run each tool so the comparison stays consistent with the Triangle Calculator.
Practical Checklist Before Relying on the Result
Before acting on the Triangle Calculator output, run a short mental checklist: inputs in the right units, direction of the result matching intuition, and magnitude plausible. Each check takes seconds and catches the most common classes of Triangle Calculator error before they reach a decision.
Putting the Result to Work
A single Triangle Calculator run usually narrows the range of plausible answers rather than settling the question. Compare the Triangle Calculator result against a benchmark or a previous run, and ask what would have to change for the answer to flip a decision.
Sensitivity to Inputs
Some inputs move the Triangle Calculator result more than others; changing each by a small amount shows which ones matter. Spend the effort on the high-impact Triangle Calculator inputs and treat the low-impact ones as approximate.
A Note on Stale Inputs
A calculation is only as fresh as the inputs that feed it, so note the date the Triangle Calculator inputs were last refreshed. A six-month-old Triangle Calculator result can be as wrong as a wrong calculation when the underlying values have moved on.