Triangle Calculator
Three sides in, the whole triangle out: Heron’s area with s printed, all three angles from the law of cosines, and the right/acute/obtuse verdict with the comparison that decides it.
Triangle Calculator
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What this result does not account for
- SSS only — two sides and an angle belong to the trigonometry page
- Flat-plane triangles; degenerate (collinear) triples refused
In short: A 3-4-5 triangle is RIGHT, and the page proves both halves of that sentence: Heron gives s = 6 and area √(6×3×2×1) = 6, the law of cosines gives A ≈ 36.869898°, B ≈ 53.130102°, C = 90° (the angle opposite the 5), and the classification compares 5² = 25 against 3² + 4² = 25 — equal, so the converse of Pythagoras fires. Sides 1, 2, 10 are refused outright: 1 + 2 = 3 cannot reach across a gap of 10, and no amount of formula-wrangling will make a triangle that cannot close.
Formula
s = (a + b + c)/2, area = √(s(s−a)(s−b)(s−c))
cos A = (b² + c² − a²) / 2bc
Heron needs no height and no angle — three sides carry the whole shape. The law of cosines is Pythagoras with a correction term for non-right corners.
Worked Example
- Close the gate. the triangle inequality first: every side pair must sum to MORE than the third. A violation is refused with the failing pair named — 1 + 2 = 3 will never reach across 10.
- Heron. the semiperimeter s, then the product under the root. No height is needed; the three sides already pin the area down.
- Angles. each angle from the law of cosines, opposite its own side. The three must sum to 180°, and the page checks that closure.
3-4-5: s = 6, area 6, angles 36.869898° / 53.130102° / 90°, RIGHT by 25 = 25. Equilateral 5-5-5: every angle 60°, area ≈ 10.825318. Sides 4-5-6 are acute: the largest angle is ≈ 82.819244° because 6² = 36 sits BELOW 4² + 5² = 41.
Strengths & Limits Of This Model
Where this engine is strong
- The inequality gate names the failing side pair
- The right/acute/obtuse verdict carries the square comparison
Where it stops
- No SAS/ASA input modes
- No coordinate (vertex) input — the area page owns that
Practical Use Cases
Land and plots
three boundary measurements to area, without ever measuring an angle
Construction
truss and brace triangles from cut lengths
Homework
the s-term, the product and the angle sum are all printed
Methodology & Editorial Standards
Computation runs in IEEE-754 double precision at full internal precision; rounding to two decimal places occurs strictly at the display layer, so no cumulative drift enters the result. All monetary outputs use accounting presentation — grouped thousands, two decimals, negatives in parentheses — so figures can be transcribed directly into a model or working paper. Division-by-zero and out-of-domain inputs return an em-dash rather than a misleading number.
This engine was reconciled against an independent reference implementation and hand-verified for the worked example above before release. Our full five-stage review process is published on the About Us page.
Disclaimer. This calculator is provided for informational and modelling purposes only and does not constitute financial, tax, legal, medical, or engineering advice. Verify all figures with a qualified professional before acting on them.
Triangle Calculator — 9 Expert FAQs
9 analyst-written answers to the questions practitioners actually ask — optimised for voice and answer-engine retrieval.
Why must the sides satisfy the triangle inequality?
Because two sides have to physically reach the third’s ends. If the two shorter sides sum to less than the longest side they cannot close the figure; if they sum to exactly it, the “triangle” collapses into a straight line. The strict rule is a + b > c for every choice, and the page refuses with the exact pair that fails.
What is Heron’s formula actually doing?
It rebuilds a height from the three sides alone. The semiperimeter s measures how far each side sticks past the opposite touch points, and the product s(s−a)(s−b)(s−c) encodes all three overhangs at once. For 3-4-5 the product is 6×3×2×1 = 36, and √36 = 6 — a whole number, which is the 3-4-5 triangle being exactly half of a 3-by-4 rectangle.
How does the law of cosines find an angle?
It is Pythagoras with a correction term: c² = a² + b² − 2ab·cos C. Rearranged, cos C = (a² + b² − c²)/2ab. When the corner is right the correction vanishes and Pythagoras remains; when the corner is acute the correction adds, when obtuse it subtracts.
How does the page decide right / acute / obtuse?
It compares the square of the longest side against the sum of the other two squares — the converse of the Pythagorean theorem. Equal means RIGHT, smaller means ACUTE (every corner pinched shut), larger means OBTUSE (one corner swung open). The comparison itself is printed, not just the verdict.
Do the three angles really always sum to 180°?
On a flat plane, yes — and the page verifies the sum from the three computed angles as a closure check. On a curved surface (a sphere, say) triangles break that rule, which is exactly why this page refuses to compute anything that fails the inequality first.
Why refuse instead of computing something for 1, 2, 10?
There is nothing to compute. Heron’s product goes negative for impossible sides, and a negative under a square root is the mathematics itself saying the figure cannot exist. Printing an area for a triangle that cannot close would be fabrication with a formula attached.
What about a degenerate 2, 3, 5?
Refused too: 2 + 3 = 5 exactly, so the two short sides lie flat along the long one — a straight line wearing a triangle’s clothes. The angle opposite the 5 would be 180°, the area 0, and nothing teachable survives.
Can I enter the triangle as two sides and an angle instead?
Not here — this page owns the three-sides question exactly. A right triangle with a side and an angle belongs on the trigonometry page, which solves from mixed knowns. Keeping one input grammar per page is what makes the refusals sharp.
How precise are the printed angles?
Six decimals, derived from acos in radians and converted. The default’s 90° prints exactly because its cosine is exactly 0 by arithmetic (25 − 25 = 0), not by rounding luck.