Calculate all properties of any triangle — sides, angles, area, and perimeter. Enter any 3 values using SSS, SAS, ASA, or SSA method — including the ambiguous SSA case, which can have zero, one, or two valid solutions.
enter triangle values
Method
Side a
Side b
Side c (SSS)
Angle A (°)
Angle B (°)
Result (click to copy)
Picking the Right Method
Know all 3 sides?
Use SSS. This is the most common case for a triangle you've physically measured — three side lengths fully determine a unique triangle.
Know 2 sides and the angle between them?
Use SAS. The included angle is the one formed where the two known sides meet — a unique triangle always results.
Know 2 sides and an angle NOT between them?
Use SSA — but be aware this is the one case that can have zero, one, or two valid triangles for the same inputs. The calculator will show you all valid solutions.
How to Use the Triangle Calculator
Select your known values — choose SSS (3 sides), SAS (2 sides + included angle), or ASA (2 angles + included side) based on what you know.
Enter the known values — sides in any consistent unit (cm, m, inches), angles in degrees.
Click Calculate — the solver finds all remaining sides, angles, area, and perimeter.
Read the full solution — see all triangle properties including type classification (equilateral, isosceles, scalene).
Verify the triangle inequality — a valid triangle requires each side to be less than the sum of the other two sides.
📐 Triangle inequality: For sides a, b, c: a + b > c, b + c > a, a + c > b. If any condition fails, no triangle exists. Example: sides 2, 3, 6 cannot form a triangle because 2 + 3 = 5 < 6. Always check this before solving — it prevents wasted calculation on impossible triangles.
Understanding Triangle Solving Methods
📐 Law of Cosines
For SSS: cos(A) = (b²+c²−a²)/2bc. Finds any angle from all three sides. For SAS: a² = b²+c²−2bc·cos(A). Finds the third side from two sides and the included angle. Generalises the Pythagorean theorem — when A=90°, cos(90°)=0 and the formula reduces to a²=b²+c².
📏 Law of Sines
a/sin(A) = b/sin(B) = c/sin(C). Ratio of each side to the sine of its opposite angle is constant for any triangle. Used for ASA and AAS cases. The ambiguous case (SSA): may produce 0, 1, or 2 valid triangles depending on whether the given side is long enough to reach the opposite angle.
📊 Heron's Formula
Area = √(s(s−a)(s−b)(s−c)) where s = (a+b+c)/2 (semi-perimeter). Calculates area from three sides alone — no height needed. Named after Hero of Alexandria (10–70 AD). The most general area formula, applicable to any triangle given all three sides.
🔺 Triangle Types
By sides: Equilateral (all sides equal, all angles 60°). Isosceles (two sides equal, two base angles equal). Scalene (all sides different). By angles: Acute (all angles < 90°). Right (one angle = 90°). Obtuse (one angle > 90°). These classifications affect which solving methods apply and what properties hold.
📈 Right Triangle Shortcuts
For right triangles, Pythagorean theorem (a²+b²=c²) and basic trigonometry (sin, cos, tan) are simpler than the general laws. SOH-CAH-TOA: sin(θ)=opposite/hypotenuse, cos(θ)=adjacent/hypotenuse, tan(θ)=opposite/adjacent. Pythagorean triples: 3-4-5, 5-12-13, 8-15-17 — recognising these speeds up calculations.
🌐 Real Applications
Surveying and navigation (triangulation to find distances). Architecture (roof structures, load analysis). Engineering (force vector decomposition). GPS positioning (trilateration from satellite distances). Computer graphics (all 3D objects are triangles). Astronomy (stellar distance measurement via parallax triangles).
Triangle Geometry in the Real World
Triangulation for distance measurement
Surveyors use triangulation to measure distances without physically reaching remote points. From two known positions (baseline), measure angles to a target point. The triangle formed by the baseline and two sight lines allows calculating the distance to the target using the Law of Sines. This technique measures heights of mountains, widths of rivers, and distances to stars — all from geometry alone. GPS trilateration uses three-dimensional versions of the same principle.
Structural triangles in engineering
Triangles are the only polygon that maintains its shape under load — other polygons deform when force is applied. This is why trusses, bridges, and towers are built from triangular frameworks. The Eiffel Tower, suspension bridge cables, and geodesic domes all rely on triangular stability. Engineers calculate forces in each triangle member using the Law of Cosines and Sines to ensure structures can bear their design loads.
Computer graphics and triangle meshes
All 3D graphics — every game, movie, and animation — represents surfaces as triangle meshes. Triangles are chosen because: any three non-collinear points define a unique plane (ensuring flat faces), triangles are always convex (simplifying rendering algorithms), and triangle calculation is extremely efficient. A modern game renders millions of triangles per frame using the triangle calculations this tool performs.
📐 Triangle solving decision chart: Know 3 sides (SSS)? → Law of Cosines for all angles. Know 2 sides + included angle (SAS)? → Law of Cosines for third side, then Law of Sines. Know 2 angles + any side (AAS/ASA)? → Third angle = 180°−A−B, then Law of Sines for sides. Know 2 sides + non-included angle (SSA)? → Check ambiguous case — may have 0, 1, or 2 solutions.
Formulas Used
Law of Cosines: c² = a² + b² − 2ab·cos(C) · Law of Sines: a/sin(A) = b/sin(B) = c/sin(C) · Area (Heron's): √(s(s−a)(s−b)(s−c)), where s = perimeter/2
Worked example (SSS)
Sides 3, 4, 5 (a classic right triangle): s = (3+4+5)/2 = 6. Area = √(6×3×2×1) = √36 = 6. Angle C (opposite side 5) = arccos((3²+4²−5²)/(2×3×4)) = arccos(0) = 90° — confirming this is a right triangle.
Worked example (SSA — ambiguous case)
a=7, b=10, A=40°: sin(B) = 10×sin(40°)/7 = 0.918, giving B = 66.66° or B = 113.34° (its supplement). Both keep A+B under 180°, so two valid triangles exist — one with C≈73.3°, one with C≈26.7°. This is why SSA is called the ambiguous case.
Avoid These Mistakes
Entering sides that can't form a triangle
The sum of any two sides must exceed the third (e.g. sides 1, 1, 10 are impossible — 1+1 doesn't reach 10). The calculator now checks this and will tell you clearly rather than showing a broken result.
Assuming SSA always gives one answer
Unlike SSS, SAS, and ASA, SSA can have zero, one, or two valid triangles for the same three inputs — check which case applies before assuming your answer is unique.
Mixing up which angle is "included"
SAS requires the angle between the two given sides. If your known angle is opposite one of the sides instead, that's SSA, not SAS — using the wrong method's formula gives a wrong answer.
Forgetting angles must sum to 180°
For ASA, if your two given angles already sum to 180° or more, no valid triangle exists — there's no room left for the third angle.
Frequently Asked Questions
What is SSS, SAS, ASA, and SSA?
SSS: you know all 3 sides. SAS: you know 2 sides and the included angle. ASA: you know 2 angles and the included side. SSS, SAS, and ASA each fully define a unique triangle. SSA (2 sides and a non-included angle) is the exception — it can have zero, one, or two valid triangles for the same inputs, known as the ambiguous case.
How do I find the area of a triangle?
Using Heron's formula: Area = √(s(s-a)(s-b)(s-c)) where s = (a+b+c)/2. Alternatively: Area = ½ × base × height. This calculator uses Heron's formula which works from side lengths alone.
References
📐 Law of Sines & Cosines
Standard trigonometric identities taught in most high school and college trigonometry courses — this calculator's SSS, SAS, ASA, and SSA methods all derive from these two laws.
📄 Heron of Alexandria
Heron's area formula, used here, dates to antiquity and remains the standard method for computing triangle area from three side lengths alone, without needing to know any angle.