Triangle Calculator – Right, SSS, SAS, 30-60-90 & Angles

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Triangle Calculator

Solve any triangle: right triangles with Pythagoras and trig, 30-60-90 special ratios, and general triangles from SSS, SAS, ASA or AAS using the law of sines and cosines. Every result comes with a drawn diagram and step-by-step math. Free, instant, private.

✓ Right + General Triangles · ✓ Drawn Diagrams · ✓ Angles, Sides & Area

🔧 JS ⏳ · Input — · Result — · Engine ⏳

Enter any two values — leave the rest blank. Angles in degrees.

Side a (opposite α)

Side b (opposite β)

Hypotenuse c

Angle α (°)

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Area

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Perimeter

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Angle β

SSS — three sides SAS — two sides + included angle ASA — two angles + included side AAS — two angles + non-included side

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Area

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Perimeter

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Largest angle

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Type

📋 All Sides & Angles

Enter any one side of a 30-60-90 triangle

side = short leg (x) side = long leg (x√3) side = hypotenuse (2x)

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Short leg

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Long leg

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Hypotenuse

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Area

Ratio: short = x · long = x√3 · hypotenuse = 2x — angles fixed at 30°, 60°, 90°.

Right Triangle Calculator — Pythagoras & Trig

A right triangle has one 90° angle; the side opposite it is the hypotenuse, always the longest. Enter any two values in the first tab and everything else solves:

• Pythagorean theorem: a² + b² = c² — legs to hypotenuse (c = √(a²+b²)) or backwards (a = √(c²−b²)).
• Trigonometry: sin(α) = a/c · cos(α) = b/c · tan(α) = a/b — and inverses (arcsin, arccos, arctan) from an angle to sides.
• Area: ½ × a × b — legs are the base and height.

Example: legs 3 and 4 → hypotenuse 5, angles ≈ 36.87° and 53.13°, area 6. Try it above — the drawn diagram and step-by-step panel update instantly.

The 30-60-90 Triangle — Fixed Ratio

short = x   ·   long = x√3  ≈ 1.732x   ·   hypotenuse = 2x

Memorize one side and you know them all: a 10 cm short leg gives a ≈17.32 cm long leg and a 20 cm hypotenuse. These appear constantly on tests because the ratios are exact — the calculator applies √3 at full precision so rounding never bites you. The 30° angle faces the short leg, the 60° faces the long leg, the 90° faces the hypotenuse.

Solving Any Triangle — SSS, SAS, ASA, AAS

• SSS (three sides): Law of Cosines finds the largest angle first: cos(C) = (a² + b² − c²) ÷ (2ab); then Law of Sines or angle sum for the rest; area via Heron's formula: s = (a+b+c)/2, Area = √(s(s−a)(s−b)(s−c)).
• SAS (two sides + angle between): Area = ½·a·b·sin(C) directly; the third side comes from Law of Cosines; remaining angles from Law of Sines.
• ASA / AAS (two angles + a side): third angle = 180° − both; then Law of Sines: a = b·sin(A) ÷ sin(B).

Each mode validates the triangle inequality (any two sides must exceed the third) and the 180° angle sum — impossible triangles are rejected with an explanation instead of returning nonsense.

Why Use This Triangle Angle Calculator?

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Four Solver Modes

Right, SSS, SAS, ASA and AAS — every triangle scenario covered in one page.

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Drawn Diagram

The solved triangle renders to scale on canvas — see it, don't imagine it.

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Steps Shown

Which law ran, with which numbers — homework-ready working out.

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Impossible-Triangle Guard

Triangle inequality and 180° sum enforced — nonsense inputs get clear messages.

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100% Private

Pure browser math — nothing sent, stored, or tracked.

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Instant & Offline

Solves on every keystroke — perfect for checking homework at speed.

Popular Use Cases

• 🎓 Geometry and trigonometry homework — verify working, not just answers
• 🏗️ Construction and carpentry — rafter angles, roof pitches, bracing lengths
• 🧭 Navigation — bearing distances forming triangles on maps
• ⚙️ Engineering layouts — bracket angles and support lengths
• 🎨 Design and art — exact proportions for triangular compositions
• 📐 Quick checks — "is 5-12-13 right?" and one-side 30-60-90 lookups

Pro Tips for Triangle Problems

• Right triangle? Two knowns is enough: any two of a, b, c, α fully determine the rest — Pythagoras handles sides, trig handles angles.
• SSS: solve the biggest angle first: Law of Cosines on the longest side avoids ambiguous arc-cosine cases and always gives a valid angle between 0° and 180°.
• Spot special triangles: 3-4-5 family (right), 30-60-90 (x, x√3, 2x), 45-45-90 (x, x, x√2). Recognizing them saves all the trig work.
• Check with the angle sum: solved angles must total 180°. The general solver does this internally; a quick mental check catches typos in manual work.
• Keep units consistent: mixing meters and centimeters across sides silently breaks area and perimeter. Convert first, then solve.

Frequently Asked Questions

Square both legs, add them, take the square root: c = √(a² + b²). Legs of 3 and 4 give √(9+16) = √25 = 5. In the Right Triangle tab, enter just the two legs — the hypotenuse, angles, area and perimeter all appear at once.

Label the longest side c and test a² + b² = c². If it holds exactly, it's right. 5-12-13 works (25+144=169); 4-5-6 doesn't (16+25=41 ≠ 36). Enter any three sides in the general SSS mode and it tells you the triangle type directly.

Its sides follow the fixed ratio 1 : √3 : 2. Know any one side and multiply: short leg = x, long leg = x√3 ≈ 1.732x, hypotenuse = 2x. The 30-60-90 tab applies these exactly — enter whichever side you have and pick which one it is.

Yes — that's SSS mode. It uses the Law of Cosines for the angles and Heron's formula for the area, after validating the triangle inequality. Impossible sets like 1-2-10 are rejected with an explanation.

SAS is two sides with the angle between them — solved with the Law of Cosines. ASA is two angles with the side between them — solved via the third angle and the Law of Sines. AAS (side not between the angles) solves the same way as ASA. All three are supported.

Three ways depending on what you know: ½ × base × height for right triangles (the legs are base and height); ½·a·b·sin(C) for SAS; and Heron's formula √(s(s−a)(s−b)(s−c)) for SSS. The calculator picks the right one for your inputs automatically.

The triangle inequality: any two sides must add up to more than the third. 2 + 3 = 5 fails against a side of 5 — the sides can't close. The tool detects this and explains which combination fails instead of returning a broken result.

No. All solving is JavaScript in your browser — sides, angles and results never leave your device. Refreshing the page clears everything.