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Triangle Solver – Solve Any Triangle SSS SAS ASA AAS

Solve any triangle completely using SSS, SAS, ASA, or AAS methods with our free online triangle solver. Find all missing sides, angles, and area with step-by-step trigonometric solutions.

SSS (Side-Side-Side): You know all three sides of the triangle.

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Understanding Triangle Solving

Solving a triangle means finding all unknown sides and angles when you're given partial information. This is one of the most practical applications of trigonometry, used in surveying, navigation, engineering, and physics.

The key insight is that triangles are rigid shapes. If you know three pieces of information about a triangle (with at least one being a side length), the entire triangle is determined. There's only one possible triangle that fits those measurements.

The four main methods are SSS (three sides), SAS (two sides and the angle between them), ASA (two angles and the side between them), and AAS (two angles and any side). Each method uses specific trigonometric laws to find the missing pieces.

Triangle Solving Formulas

Law of Cosines

a² = b² + c² - 2bc × cos(A)
b² = a² + c² - 2ac × cos(B)
c² = a² + b² - 2ab × cos(C)

Use for SSS and SAS. This generalizes the Pythagorean theorem – when angle C is 90°, cos(90°) = 0, and you get a² + b² = c².

Law of Sines

a/sin(A) = b/sin(B) = c/sin(C)

Use for ASA and AAS. This relates each side to the sine of its opposite angle. The ratio is the same for all three sides.

Heron's Formula (Area)

s = (a + b + c) / 2
Area = √[s(s-a)(s-b)(s-c)]

Find area when you know all three sides. Named after Hero of Alexandria, who proved it around 60 AD.

Trig Area Formula

Area = ½ × b × c × sin(A)
Area = ½ × a × c × sin(B)
Area = ½ × a × b × sin(C)

Use when you know two sides and the included angle. This is often easier than Heron's formula for SAS problems.

Worked Examples

SSS Example: Sides 5, 7, 8
Given: a=5, b=7, c=8
cos(A) = (49+64-25)/(2×7×8) = 88/112 = 0.7857
A = arccos(0.7857) = 38.21°
cos(B) = (25+64-49)/(2×5×8) = 40/80 = 0.5
B = arccos(0.5) = 60°
C = 180° - 38.21° - 60° = 81.79°
Area = √[10×5×3×2] = √300 = 17.32
SAS Example: b=6, A=45°, c=8
Given: b=6, A=45°, c=8
a² = 36 + 64 - 96×cos(45°) = 100 - 67.88 = 32.12
a = √32.12 = 5.67
sin(B)/6 = sin(45°)/5.67
B = arcsin(6×0.707/5.67) = 47.06°
C = 180° - 45° - 47.06° = 87.94°
Area = ½×6×8×sin(45°) = 16.97
ASA Example: B=30°, c=10, A=60°
Given: B=30°, c=10, A=60°
C = 180° - 30° - 60° = 90° (right triangle!)
a/sin(60°) = 10/sin(90°)
a = 10 × 0.866/1 = 8.66
b/sin(30°) = 10/sin(90°)
b = 10 × 0.5/1 = 5
Area = ½×8.66×5 = 21.65
AAS Example: A=40°, B=70°, c=12
Given: A=40°, B=70°, c=12
C = 180° - 40° - 70° = 70°
Since B = C, this is isosceles! b = c = 12
a/sin(40°) = 12/sin(70°)
a = 12 × 0.643/0.940 = 8.21
Area = ½×8.21×12×sin(70°) = 46.27
Real-World: Surveying a Triangular Plot
A surveyor measures two sides of a plot: 150m and 200m
The angle between them is 73°
Third side: c² = 150² + 200² - 2×150×200×cos(73°)
c² = 22500 + 40000 - 60000×0.292 = 44960
c = 212m (fencing needed)
Area = ½×150×200×sin(73°) = 14,340 m² (land area)

Quick Fact: The History of Triangle Solving

Ancient Greek astronomer Hipparchus (190-120 BC) created the first trigonometric table to solve triangles for astronomical calculations. The Law of Sines was known to Islamic mathematicians by the 10th century. Persian mathematician Al-Kashi (1380-1429) explicitly stated the Law of Cosines, which is still called "Al-Kashi's Theorem" in some countries. French mathematician François Viète (1540-1603) systematized triangle solving methods that we still use today.

Frequently Asked Questions

What does it mean to "solve" a triangle?

Solving a triangle means finding all unknown sides and angles. If you're given three pieces of information (with at least one side), you can determine everything else about the triangle using trigonometric laws.

Can I solve a triangle with only angles (AAA)?

No. Knowing all three angles only tells you the shape, not the size. You need at least one side length to determine the scale. Triangles with the same angles but different sizes are called similar triangles.

What is the ambiguous case (SSA)?

SSA (two sides and a non-included angle) can produce zero, one, or two possible triangles. This calculator doesn't handle SSA because it requires checking multiple cases. Use Law of Sines carefully when you encounter this situation.

Why do my angles not add up to exactly 180°?

Rounding. The calculator shows rounded values for display, but uses full precision internally. If you add the displayed angles and get 179.99° or 180.01°, that's just rounding error. The actual angles do sum to exactly 180°.

When should I use Law of Sines vs Law of Cosines?

Law of Cosines works for SSS and SAS. Law of Sines works for ASA and AAS. Think of it this way: Law of Cosines finds angles from sides (or a side from two sides and an angle). Law of Sines finds sides from angles (or angles from sides).

Can this calculator handle obtuse triangles?

Yes. The Law of Cosines naturally handles obtuse angles (angles greater than 90°). The arccos function returns angles in the range 0° to 180°, covering both acute and obtuse cases.

How accurate are the results?

The calculator uses double-precision floating-point arithmetic, giving about 15-16 significant digits of accuracy. Results are displayed rounded to 4 decimal places for readability, but internal calculations maintain full precision.

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