Triangle Calculator — SSS, SAS, ASA, AAS Solver
Enter any 3 measurements — at least one side — and we'll find every missing side, angle, the area and the perimeter in one click. Works as an SSS, SAS, ASA, AAS or SSA calculator for right, isosceles, scalene, equilateral and obtuse oblique triangles using the laws of sines and cosines.
How to Use This Triangle Solver
This is a full trig triangle calculator — give it any 3 known values and it will compute every missing measurement using the appropriate trigonometric law. It handles all five classical cases (SSS, SAS, ASA, AAS, SSA), warns you about the ambiguous SSA case, and works equally well for right triangles, isosceles triangles and obtuse oblique triangles.
- Decide what you know. Three sides? Two sides plus an angle? Two angles plus a side? You need exactly 3 values, and at least one of them must be a side length — three angles alone never define a unique triangle (similar triangles can have different sizes).
- Enter the values into the matching fields. Sides are labelled a, b, c. Angles are A, B, C in degrees, with A opposite side a, B opposite b, C opposite c.
- Click Solve. The calculator reports all six measurements plus the area (using Heron's formula or the SAS area formula, whichever is more accurate) and the perimeter.
- Try the 3-4-5 example button to see how a classic right triangle gets solved instantly.
How Triangles Are Solved (The 5 Classic Cases)
To uniquely solve a triangle you need 3 independent measurements, and at least one of them must be a side. The five named cases:
| Case | What You Know | How to Solve |
|---|---|---|
| SSS | All three sides | Law of Cosines for each angle |
| SAS | Two sides + the angle between them | Law of Cosines for third side, then Law of Sines |
| ASA | Two angles + the side between them | Third angle by subtraction; Law of Sines |
| AAS | Two angles + a side not between them | Same as ASA; third angle, then Law of Sines |
| SSA | Two sides + non-included angle | "Ambiguous case" — may have 0, 1, or 2 valid solutions |
For the foundational laws, see Wikipedia: Law of Sines and Law of Cosines.
Area Formulas
| Formula | When to Use |
|---|---|
| Area = (1/2) × base × height | You know a side and its perpendicular height |
| Area = (1/2) × a × b × sin(C) | You know two sides and the included angle |
| Area = √[s(s−a)(s−b)(s−c)] | Heron's formula — all three sides, s = perimeter/2 |
AAS, ASA, SSS, SAS Calculator — Pick Your Mode
Same calculator, four modes depending on what you already know. Just leave the fields you don't know empty — the solver picks the right method.
| Mode | Fill in | Example | Method used |
|---|---|---|---|
| SSS | Sides a, b, c | a=5, b=6, c=7 | Law of cosines for each angle |
| SAS | Two sides + angle between them | a=5, b=7, C=60° | Law of cosines for third side, then sines |
| ASA | Two angles + side between them | A=45°, B=70°, c=8 | Third angle = 180−A−B; law of sines |
| AAS | Two angles + side not between | A=45°, B=70°, a=6 | Same as ASA |
| SSA | Two sides + non-included angle | a=7, b=10, A=40° | Law of sines — ambiguous case (0, 1, or 2 solutions) |
The most famous triangle in mathematics
Sides of 3, 4, and 5 form a right triangle (3² + 4² = 25 = 5²). This is the simplest Pythagorean triple and has been used by builders to lay out right angles for millennia — the ancient Egyptians used "rope stretchers" with knots at 3, 4, and 5 spacings to construct true right angles for pyramid foundations.
| Property | Value |
|---|---|
| Sides | a=3, b=4, c=5 |
| Angles | A=36.87°, B=53.13°, C=90° |
| Area | 6 (via base×height/2) |
| Perimeter | 12 |
Other notable Pythagorean triples: 5-12-13, 8-15-17, 7-24-25, 20-21-29. All are infinite families — multiply any triple by an integer (6-8-10, 9-12-15) and you get another valid right triangle.
Special Triangles to Know
- Equilateral: all sides equal, all angles 60°. Most symmetric.
- Isosceles: two sides equal, two angles equal (the angles opposite the equal sides).
- Scalene: all sides different lengths, all angles different.
- Right (90°): Pythagorean theorem applies: a² + b² = c² where c is hypotenuse.
- Obtuse: one angle greater than 90°. Area still works with Heron's formula — no special case needed.
- 30-60-90: sides in ratio 1 : √3 : 2. Common in trigonometry.
- 45-45-90 (isosceles right): sides in ratio 1 : 1 : √2.
Area Formulas for Special Triangles
Heron's formula works for any triangle (acute, right, obtuse — all the same). These shortcuts skip Heron's when you have the right inputs:
| Triangle type | Inputs known | Area formula |
|---|---|---|
| Equilateral | Side a | Area = (√3 / 4) × a² |
| Isosceles | Equal sides a, base b | Area = (b / 4) × √(4a² − b²) |
| Isosceles | Equal sides a, apex angle θ | Area = (a² / 2) × sin(θ) |
| Right triangle | Two legs a, b | Area = (a × b) / 2 |
| Obtuse / any | All three sides | Heron: √[s(s−a)(s−b)(s−c)] |
| 30-60-90 | Short leg a | Area = (a² × √3) / 2 |
| 45-45-90 | Leg a | Area = a² / 2 |
The calculator above runs Heron's automatically as soon as all three sides are filled — so for an isosceles triangle with sides 5, 5, 6 just enter a=5, b=5, c=6 and the area pops out (12 in that case). Same trick for obtuse triangles.
Where Triangle Math Is Used
- Surveying and GPS: triangulation determines position from known reference points — the same math that lets your phone pinpoint you within 5 metres
- Architecture and construction: trusses, roof angles, staircase rise/run, structural load distribution, every angle on a hip roof
- Astronomy: stellar parallax measures distance to nearby stars via triangle geometry over Earth's orbit
- Computer graphics and games: every 3D model you have ever seen — from Pixar films to video games — is built from millions of triangles
- Navigation: the law of cosines underlies the great-circle distance formula used by GPS and aviation
- Carpentry and DIY: the 3-4-5 method still gets used on building sites every day to square up wall layouts
How to find a roof angle from rise and run
You measured a roof rafter: it rises 4 m vertically over a horizontal span of 6 m. What's the pitch angle and the rafter length?
This is a right triangle with the two legs (rise 4, run 6) known. Plug in: side a = 4, side b = 6, angle C = 90°. The solver returns:
- Hypotenuse (rafter length): c ≈ 7.21 m (from √(4² + 6²))
- Pitch angle: A ≈ 33.69° (this is the roof's slope)
- Area: 12 m² (rise × run ÷ 2)
Same trick works for ramps, staircase pitches, or any rise-over-run problem — the calculator just calls them "side a" and "side b".
Frequently Asked Questions
How do I use this as an AAS, ASA or SSS calculator?
Enter the values you know in the labeled fields. For AAS or ASA, fill any 2 angles plus 1 side. For SSS, fill all 3 sides and leave the angles blank. For SAS, fill 2 sides plus the included angle. For SSA, fill 2 sides and the non-included angle. The calculator picks the right method automatically.
How do I calculate the third side of a triangle?
Depends on what you know. With 2 sides plus the included angle (SAS), use the law of cosines: c² = a² + b² − 2ab·cos(C). With 2 sides plus a non-included angle (SSA), use the law of sines. For right triangles with the two legs known, c = √(a² + b²) (Pythagoras).
How do I find the area of an isosceles triangle?
With two equal sides a and base b: Area = (b/4) × √(4a² − b²). Or just enter a, a, b in the calculator above — Heron's formula handles it automatically. For an isosceles triangle with apex angle θ and equal sides a: Area = (a²/2) × sin(θ).
How do I find the area of an obtuse triangle?
Same formulas as any other triangle. Heron's formula works regardless of whether the triangle is obtuse, acute or right: Area = √[s(s−a)(s−b)(s−c)] where s = (a+b+c)/2. With 2 sides and the included angle: Area = (1/2)·a·b·sin(C).
What's the area of a triangle from SSS?
Heron's formula. Compute the semi-perimeter s = (a+b+c)/2, then Area = √[s(s−a)(s−b)(s−c)]. Example: sides 5, 6, 7 → s = 9 → Area = √(9·4·3·2) = √216 ≈ 14.70.
How does the 3-4-5 triangle calculator work?
The 3-4-5 triangle is the simplest Pythagorean triple: 3² + 4² = 5². It's a right triangle with angles 36.87°, 53.13°, 90°. Enter a=3, b=4, c=5 and the solver returns those angles plus area = 6 and perimeter = 12. Carpenters still use it to lay out true right angles on building sites.
What is a trig (trigonometry) triangle calculator?
"Trig triangle" is a casual name for any triangle whose missing parts can be found using trigonometry — specifically the law of sines and the law of cosines. This tool is exactly that: a trigonometry-based solver for non-right triangles as well as right ones. The Pythagorean theorem is just the special case where one angle is 90°.
How do I measure a triangle without a protractor?
If you can only measure lengths, measure all three sides accurately and enter them as a, b, c. The calculator will derive all angles via the law of cosines. This is exactly how surveyors and CNC machinists work when angle-measuring tools aren't available.
Why does the SSA case sometimes give two answers?
When you know two sides and an angle that is not between them, there can be 0, 1, or 2 valid triangles. This is called the ambiguous case. The calculator reports the most likely solution; if you suspect a second valid triangle, try entering the supplementary angle (180° − A) and compare.
What's the difference between a triangle solver and a Pythagorean calculator?
A Pythagorean calculator handles only right triangles (one 90° angle). A full triangle solver — like this one — handles right, acute and obtuse triangles using the more general laws of sines and cosines. The Pythagorean theorem is a special case of the law of cosines when C = 90° (because cos 90° = 0, the law collapses to a² + b² = c²).
Can three angles alone define a triangle?
No. Three angles describe the shape but not the size — there are infinitely many similar triangles with the same angles. You must always supply at least one side length. The calculator will refuse to solve if you give only angles.
The Math, Briefly
Two laws cover every triangle problem:
| Law | Formula | When to Use |
|---|---|---|
| Law of Sines | a / sin A = b / sin B = c / sin C | You know an angle and the side opposite it |
| Law of Cosines | c² = a² + b² − 2ab·cos(C) | SSS (find any angle) or SAS (find third side) |
The Pythagorean theorem is what you get when C = 90°: cos 90° = 0, so the formula collapses to c² = a² + b². See Wikipedia: Solution of Triangles for the full mathematical treatment.