Why does the SSA case sometimes have two solutions?

When you know two sides and an angle that is NOT between them (side a, side b, and angle A opposite side a), the law of sines gives you sin B = b·sin A / a. The arcsine of that value has two angles in (0°, 180°) — an acute one and its supplement — and both can sometimes form a valid triangle. Picture swinging side a like a compass arm from the end of side b: if it is long enough it crosses the base line in two places, giving two triangles. The calculator checks both: if A + B stays under 180° for each candidate, that triangle is real, so you can get one solution, two solutions, or none at all.

When do I use the law of sines versus the law of cosines?

Use the law of cosines when you have three sides (SSS) or two sides and the angle between them (SAS) — it is the only one that works without a known side-angle pair to start from. Use the law of sines when you already have a matched side-and-its-opposite-angle pair plus one more piece (ASA, AAS, SSA). A practical rule: cosines turns sides into angles and closes the gap when you have no ratio yet; sines extends an existing ratio to find the rest. This tool picks the right one automatically based on the case you select.

Should I use Heron's formula or ½·a·b·sin C for the area?

They give the same answer, but each fits different inputs. Heron's formula, area = √(s(s−a)(s−b)(s−c)) with s = (a+b+c)/2, needs all three sides, so it is the natural choice once a triangle is fully solved or when you started from SSS. The ½·a·b·sin C form needs two sides and the angle between them, so it is exact and direct for the SAS case before you have even found the third side. This calculator reports area from the solved sides via Heron, which matches the SAS formula to the last digit.

What counts as an invalid triangle?

Two things make a triangle impossible. First, the triangle inequality: each side must be shorter than the sum of the other two, so 1, 1, 5 cannot close into a triangle. Second, the angle sum: the three interior angles must total exactly 180° (π radians), so if two known angles already add to 180° or more, no triangle is left. The calculator flags both with a specific message, plus the SSA "side too short to reach" case, instead of returning a misleading result.

Can I work in radians instead of degrees?

Yes. The angle-unit toggle switches every angle input and output between degrees and radians, and your choice is remembered for next visit. Internally the solver always works in radians (that is what JavaScript's Math.sin and Math.acos expect) and converts only at the edges, so you get the same precision either way. Side lengths are unitless — feet, metres, or pixels all work, since the triangle math only cares about ratios.

Finish a geometry homework problem that gives you two sides and an angle

The worksheet says: "In triangle ABC, a = 8, b = 6, and angle C between them is 40°. Find side c and the area." That is a textbook SAS case. Select SAS, type a=8, b=6, C=40, and the solver returns c via the law of cosines, the other two angles, and the area from ½·a·b·sin C in one screen. You get both the answer and the intermediate angles, so you can check each step of your own working rather than just the final number.

Survey a plot of land you can only measure two ways

You walked off two edges of a triangular field — 42 m and 31 m — and measured the angle where they meet with a phone compass: 78°. You need the third edge and the area to estimate fencing and seed. Enter the SAS case (a=42, b=31, C=78). The law of cosines gives the missing edge directly, and the area tells you how much ground you are covering. No need to physically pace the awkward third side across a ditch.

Understand why your SSA problem has "two answers"

A trig class hits the ambiguous case and it feels like a trick. Set the calculator to SSA, enter a=7 opposite A=30°, with b=10. It shows two triangles: one where B is acute and one where B is its supplement. The scale drawings make it concrete — same a, same b, same A, two genuinely different shapes. Seeing both plotted is what makes the "swing the compass arm" explanation finally click.

Verify a CAD or game-engine triangle before trusting it

You computed a triangle's vertices in code and want to confirm the interior angles and area match expectations. Drop the three side lengths into SSS mode. The solver reports all three angles (so you can confirm none collapsed to 0° or 180° from a floating-point bug), the area, and whether it is right-angled — useful when a shader or collision check assumes a non-degenerate triangle.

Plan a roof truss or ramp from two angles and a span

A truss spans 6 m at the bottom, with a 35° rise on the left and a 50° rise on the right. That is ASA: two angles with the side between them. Enter A=35, B=50, c=6, and the solver gives the two rafter lengths (sides a and b) and the apex angle. The to-scale drawing lets you eyeball whether the pitch looks right before you cut any timber.

Triangle Calculator — Solve SSS / SAS / ASA / AAS / SSA, Area, Type

Triangle solver — SSS, SAS, ASA, AAS, and the ambiguous SSA case via law of sines + law of cosines. All sides, angles, area, perimeter, type, drawn to scale.

Runs locally
Category Calculator
Best for Getting a realistic range before a purchase, plan, workout, or schedule decision.

Triangle Calculator

Enter any valid combination — three sides, two sides and the included angle, two angles and a side, or the ambiguous side-side-angle case — and get every remaining side, angle, the area, perimeter, and triangle type, plotted to scale.

Angle unit:

Known combination

All three sides a, b, c.

Side aSide bSide c

Results

Sides: a = 3, b = 4, c = 5

Angles: A = 36.87°, B = 53.13°, C = 90°

Area: 6

Perimeter: 12

Type: right · scalene

Every law-of-sines and law-of-cosines step runs in your browser. Nothing is uploaded.

What this tool does

A complete triangle solver: give it any valid combination of sides and angles and it returns the whole triangle, not just the one piece you were missing. Pick the case that matches what you know — SSS (three sides), SAS (two sides and the included angle), ASA (two angles and the side between them), AAS (two angles and a side opposite one of them), or SSA (two sides and a non-included angle, the famous "ambiguous case"). Behind the scenes it uses the law of cosines to turn three sides into angles or two sides and the included angle into the third side, and the law of sines to chase the remaining parts. For every result it also computes the area (Heron's formula from three sides, or ½·a·b·sin C straight from the SAS data), the perimeter, and classifies the triangle two ways: by its largest angle (acute, right, or obtuse) and by side equality (equilateral, isosceles, or scalene). A dynamic SVG draws the triangle to scale with the three vertices labelled, so you can sanity-check that the numbers describe the shape you expected. When you feed it an SSA case that genuinely has two answers, it shows both triangles side by side instead of silently picking one. Invalid inputs — sides that break the triangle inequality, angles that already sum past 180°, or an SSA setup where the side is too short to reach — get a plain-language explanation rather than a wrong number. Every calculation runs in your browser; nothing is uploaded.

Tool details

Input: Text + Numbers; The page exposes text boxes, numeric controls, file pickers, or structured inputs depending on the tool.
Output: Live result + Preview; The result area focuses on usable output, with copy, download, or preview actions when supported.
Privacy: Browser-side processing; The main tool logic does not call an external API, so inputs normally stay in the current tab.
Save / share: Shareable URL state; Key settings are encoded in the URL so another person can reopen the same setup.
Performance budget: Initial JS <= 14 KB; No WASM budget is declared, keeping the tool quick to open on mobile.
Best fit: Calculator · Student; Category and role tags drive related tools, internal links, and quick fit checks.

How to use

1. Input

Paste or drop your content into the tool panel.
2. Process

Click the button. All processing is local in your browser.
3. Copy / Download

Copy the result or download to disk in one click.

How Triangle Calculator fits into your work

Use it for fast estimates, comparisons, and planning numbers before you make the final call.

Calculation jobs

Getting a realistic range before a purchase, plan, workout, or schedule decision.
Comparing scenarios by changing one input at a time.
Turning rough assumptions into a number you can discuss.

Calculation checks

Double-check units, dates, rates, and rounding assumptions.
Treat health, finance, tax, and legal outputs as planning aids, not professional advice.
Save the inputs that produced an important result so you can reproduce it later.

Good next steps

These links move the current task into a more complete workflow.

Real-world use cases

Finish a geometry homework problem that gives you two sides and an angle
The worksheet says: "In triangle ABC, a = 8, b = 6, and angle C between them is 40°. Find side c and the area." That is a textbook SAS case. Select SAS, type a=8, b=6, C=40, and the solver returns c via the law of cosines, the other two angles, and the area from ½·a·b·sin C in one screen. You get both the answer and the intermediate angles, so you can check each step of your own working rather than just the final number.
Survey a plot of land you can only measure two ways
You walked off two edges of a triangular field — 42 m and 31 m — and measured the angle where they meet with a phone compass: 78°. You need the third edge and the area to estimate fencing and seed. Enter the SAS case (a=42, b=31, C=78). The law of cosines gives the missing edge directly, and the area tells you how much ground you are covering. No need to physically pace the awkward third side across a ditch.
Understand why your SSA problem has "two answers"
A trig class hits the ambiguous case and it feels like a trick. Set the calculator to SSA, enter a=7 opposite A=30°, with b=10. It shows two triangles: one where B is acute and one where B is its supplement. The scale drawings make it concrete — same a, same b, same A, two genuinely different shapes. Seeing both plotted is what makes the "swing the compass arm" explanation finally click.
Verify a CAD or game-engine triangle before trusting it
You computed a triangle's vertices in code and want to confirm the interior angles and area match expectations. Drop the three side lengths into SSS mode. The solver reports all three angles (so you can confirm none collapsed to 0° or 180° from a floating-point bug), the area, and whether it is right-angled — useful when a shader or collision check assumes a non-degenerate triangle.
Plan a roof truss or ramp from two angles and a span
A truss spans 6 m at the bottom, with a 35° rise on the left and a 50° rise on the right. That is ASA: two angles with the side between them. Enter A=35, B=50, c=6, and the solver gives the two rafter lengths (sides a and b) and the apex angle. The to-scale drawing lets you eyeball whether the pitch looks right before you cut any timber.

Common pitfalls

Mislabeling which side is opposite which angle. By convention side a is opposite angle A, b opposite B, c opposite C. In SSA the known angle must be opposite one of the known sides — if you enter the angle that sits between the two sides, that is SAS, not SSA, and the answer will be wrong.
Assuming SSA always has one neat answer. It can have zero, one, or two triangles. Picking only the acute solution and ignoring the obtuse one loses a valid answer in genuinely ambiguous setups; this tool shows both so you do not silently drop a case.
Rounding each angle to a whole degree mid-calculation and then expecting the three to sum to exactly 180°. Small rounding errors compound. Keep full precision until the final display — the solver does this internally, but if you re-enter rounded values from a previous step the error stacks up.

Privacy

Every step — the law of cosines, the law of sines, Heron's formula, the angle-sum and triangle-inequality checks, the SSA ambiguity test, and the to-scale SVG drawing — runs as plain JavaScript inside your browser tab. No side length, angle, or result is ever sent to a server, logged, or analyzed, and the calculator works offline once the page has loaded. The one thing to know: the shareable URL encodes your current inputs in the query string so a "share this triangle" link reproduces the exact figure for the recipient. Those are plain geometry numbers, not secrets, so this is by design — but if for some reason you would rather not put the values in a link, just read them off the screen instead of sharing the URL.

FAQ

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Triangle Calculator — Solve SSS / SAS / ASA / AAS / SSA, Area, Type

Triangle Calculator

Results

What this tool does

Tool details

How to use

1. Input

2. Process

3. Copy / Download

How Triangle Calculator fits into your work

Calculation jobs

Calculation checks

Good next steps

Real-world use cases

Finish a geometry homework problem that gives you two sides and an angle

Survey a plot of land you can only measure two ways

Understand why your SSA problem has "two answers"

Verify a CAD or game-engine triangle before trusting it

Plan a roof truss or ramp from two angles and a span

Common pitfalls

Privacy

FAQ

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