What's the difference between SAS and SSA?

SAS means two sides plus the angle between them — always one unique solution. SSA means two sides plus an angle not between them — and that can yield 0, 1, or 2 valid triangles depending on the numbers. Order matters when you describe your inputs. Getting these mixed up is the most common source of confusion in triangle problems, so be careful which angle you're providing.

Why does my SSS input fail?

Almost always the triangle inequality — the sum of any two sides must be strictly greater than the third side. Try (3, 4, 8): 3 + 4 equals 7, which is less than 8, so no triangle exists with these sides. The two short sides literally can't meet to close the gap. If your inputs violate the inequality, the calculator catches it and tells you instead of producing a nonsense answer.

How is area computed?

Area = ½ × a × b × sin(C), where C is the angle between sides a and b. When you give the calculator three sides (SSS), it first finds an angle using the Law of Cosines, then applies the area formula. This is mathematically equivalent to Heron's formula but a bit cleaner computationally. Either way, you get the same number.

What units does the calculator use?

Unitless — the units come from whatever you input. If your sides are in meters, the area comes out in square meters. If they're in feet, area is in square feet. Just stay consistent within a single calculation. Angles are always in degrees both for input and output, since that's what most people working with triangles expect.

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Triangle Solver

Math

Solve any triangle from SSS, SAS, ASA, AAS, SSA, or right-triangle (Pythagoras) inputs. Drawn-to-scale visualization with all sides, angles, area, and perimeter.

Math

Triangle Solver

Generated on May 23, 2026

What do you know?

Know all three sides

Side a

Side b

Side c

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Triangle Solver — SSS

Area 6.000 sq · Perimeter 12.000 · Sides 3.00, 4.00, 5.00

Generated

5/23/2026

Area

6.000

sq units

Perimeter

12.000

units

Type

Right scalene

Largest angle

90.00°

Triangle (drawn to scale)

All sides & angles

Side a3.0000

Side b4.0000

Side c5.0000

Angle A36.87°

Angle B53.13°

Angle C90.00°

Sum of angles180.00°

Step-by-step calculation

Formula

Law of cosines: c² = a² + b² − 2ab·cos(C). Law of sines: a/sin(A) = b/sin(B) = c/sin(C). Area = ½·ab·sin(C).

1Mode: SSS.
2Given values produce a triangle with sides 3.00, 4.00, 5.00.
3Angles: A=36.87°, B=53.13°, C=90.00° (sum = 180.00°, must be 180°).
4Area via SAS formula: ½·3.00·4.00·sin(90.00°) = 6.0000 sq units.
5Perimeter: 3.00 + 4.00 + 5.00 = 12.0000 units.

Quick Answer

Solve any triangle from SSS, SAS, ASA, AAS, SSA, or right-triangle (Pythagoras) inputs. Drawn-to-scale visualization with all sides, angles, area, and perimeter.

Overview — Triangle Solver

Math

The Triangle Solver finds every side, angle, area, perimeter, and triangle classification (acute / right / obtuse, scalene / isosceles / equilateral) from any valid combination of three known values: SSS (three sides), SAS (two sides + included angle), ASA (two angles + included side), AAS (two angles + non-included side), SSA (two sides + non-included angle — the 'ambiguous case'), or a right triangle (Pythagoras). The result is rendered as a drawn-to-scale 2D triangle with side and angle labels, so you can verify the geometry visually. Uses the Law of Sines, Law of Cosines, and Heron's formula under the hood.

Formula

Law of cosines: c² = a² + b² − 2ab·cos(C). Law of sines: a/sin(A) = b/sin(B) = c/sin(C). Area = ½·ab·sin(C). Pythagoras (right triangle): c² = a² + b².

By Muhammad Salman SaleemUpdated: May 23, 2026MethodologyFree, no sign-upRuns in your browser — your data stays private

From the editors — what to know before you start

You're surveying a plot of land in rural Punjab, you've got two angles and a baseline distance, and you need the third side to know whether the boundary fence will fit the property line. Or you're a carpenter cutting a roof rafter — given the run and the rise, what's the hypotenuse and what's the angle? Or it's geometry homework and the question is one of those SSA puzzles that secretly has two valid answers and you've only found one. This solver handles every classical triangle configuration: SSS (three sides), SAS (two sides plus the included angle), ASA, AAS, SSA (the famously ambiguous one), and right-triangle Pythagoras. Behind the scenes it uses the Law of Sines, Law of Cosines, and Heron's formula. The drawn-to-scale 2D visualization helps catch input errors — if the triangle looks impossibly thin or doesn't close, you can spot the problem visually before chasing a number. Pro tip: always start by checking the triangle inequality. The sum of any two sides must be greater than the third. Sides 3, 4, 8 don't form a triangle (3 + 4 < 8), and SSS inputs that violate this rule are the most common 'why doesn't it work' question. For SSA problems, remember that two valid triangles can satisfy the same constraints — don't stop at the first answer if the problem hints at multiple solutions.

Pro tips most people learn the hard way

The triangle inequality is the first thing to check on any SSS input. Sum of any two sides must exceed the third. If you've got 3, 4, 8 as sides and the calculator complains, it's not broken — those numbers genuinely don't form a triangle. The two short sides can't meet across the long one.
SSA is the only triangle case that can have two valid solutions — the famous 'ambiguous case.' If you're handed two sides and an angle that's not between them, always check whether two triangles satisfy the constraints. The calculator reports both when they exist; don't accept the first solution as the only one.
Law of Cosines is just Pythagoras with an angle correction. When C = 90°, cos(C) = 0 and c² = a² + b² + (the −2ab·cos term vanishes). That single elegance is why Pythagoras is sometimes called 'a special case of the Law of Cosines' — they're the same formula, with the right triangle being the simplest case.
Heron's formula gives you area from three sides alone — no angles needed. Useful when you've got SSS data and just want area without finding any angles. A = √(s(s−a)(s−b)(s−c)) where s is the semi-perimeter. The calculator uses an equivalent formula internally, but Heron's is what gets cited in geometry textbooks.
Honest take: surveyors, civil engineers, and roofers actually use this math every day. Triangulating distances from two known points, computing roof pitches, laying out angles for foundations — all SSA, SAS, ASA work. If you're in construction or surveying, knowing your triangle cases isn't academic; it's a job skill.

?What is the Triangle Solver?

The Formula

Law of cosines: c² = a² + b² − 2ab·cos(C). Law of sines: a/sin(A) = b/sin(B) = c/sin(C). Area = ½·ab·sin(C). Pythagoras (right triangle): c² = a² + b².

The Law of Cosines generalizes Pythagoras: when C = 90°, cos(C) = 0 and the formula reduces to c² = a² + b². For any other angle, the −2ab·cos(C) term adjusts for the angle. The Law of Sines is used when you know an angle and its opposite side. The SSA case is 'ambiguous' because the geometry can sometimes form two valid triangles — when this happens, the calculator reports both. Heron's formula gives area from three sides alone: A = √(s(s−a)(s−b)(s−c)) where s = (a+b+c)/2.

Practical Examples

3-4-5 right triangle: SSS check — area 6, angles 36.87° / 53.13° / 90°.

Surveying: from two known angles and a baseline distance (ASA), compute the distance to a far point.

Carpentry: cutting a roof rafter — given run + rise, find the hypotenuse and angle.

Navigation: given two bearings to a landmark and the distance between observation points (AAS), triangulate the landmark's position.

Equilateral triangle (60-60-60): every triangle with all three angles equal is also equilateral (all sides equal).

Sum of any two sides MUST exceed the third — the triangle inequality. The calculator catches violations.

Frequently Asked Questions

When you know two sides and an angle that's not between them, sometimes two different triangles fit the same constraints — and both are geometrically valid. This calculator detects when SSA is ambiguous and reports both solutions side by side. SSA is the only case in classical trigonometry that can have multiple answers, which is why it gets singled out in textbooks as 'the ambiguous case.'

Popular Conversions

Jump to a ready-made conversion — useful for quick reference and sharing:

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Find the LCM and GCD (HCF) of two or more integers using the Euclidean algorithm — useful for simplifying fractions, finding common denominators, and scheduling repeating events.

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Premium Converters

Triangle Solver

Math

Solve any triangle from SSS, SAS, ASA, AAS, SSA, or right-triangle (Pythagoras) inputs. Drawn-to-scale visualization with all sides, angles, area, and perimeter.

Math

Triangle Solver

Generated on May 23, 2026

What do you know?

Know all three sides

Side a

Side b

Side c

Save / share— download or share to WhatsApp, Facebook & more

Premium Converters

Triangle Solver — SSS

Area 6.000 sq · Perimeter 12.000 · Sides 3.00, 4.00, 5.00

Generated

5/23/2026

Area

6.000

sq units

Perimeter

12.000

units

Type

Right scalene

Largest angle

90.00°

Triangle (drawn to scale)

All sides & angles

Side a3.0000

Side b4.0000

Side c5.0000

Angle A36.87°

Angle B53.13°

Angle C90.00°

Sum of angles180.00°

Step-by-step calculation

Formula

Law of cosines: c² = a² + b² − 2ab·cos(C). Law of sines: a/sin(A) = b/sin(B) = c/sin(C). Area = ½·ab·sin(C).

1Mode: SSS.
2Given values produce a triangle with sides 3.00, 4.00, 5.00.
3Angles: A=36.87°, B=53.13°, C=90.00° (sum = 180.00°, must be 180°).
4Area via SAS formula: ½·3.00·4.00·sin(90.00°) = 6.0000 sq units.
5Perimeter: 3.00 + 4.00 + 5.00 = 12.0000 units.

Quick Answer

Solve any triangle from SSS, SAS, ASA, AAS, SSA, or right-triangle (Pythagoras) inputs. Drawn-to-scale visualization with all sides, angles, area, and perimeter.

Overview — Triangle Solver

Math

Formula

Law of cosines: c² = a² + b² − 2ab·cos(C). Law of sines: a/sin(A) = b/sin(B) = c/sin(C). Area = ½·ab·sin(C). Pythagoras (right triangle): c² = a² + b².

By Muhammad Salman SaleemUpdated: May 23, 2026MethodologyFree, no sign-upRuns in your browser — your data stays private

From the editors — what to know before you start

Pro tips most people learn the hard way

The triangle inequality is the first thing to check on any SSS input. Sum of any two sides must exceed the third. If you've got 3, 4, 8 as sides and the calculator complains, it's not broken — those numbers genuinely don't form a triangle. The two short sides can't meet across the long one.
SSA is the only triangle case that can have two valid solutions — the famous 'ambiguous case.' If you're handed two sides and an angle that's not between them, always check whether two triangles satisfy the constraints. The calculator reports both when they exist; don't accept the first solution as the only one.
Law of Cosines is just Pythagoras with an angle correction. When C = 90°, cos(C) = 0 and c² = a² + b² + (the −2ab·cos term vanishes). That single elegance is why Pythagoras is sometimes called 'a special case of the Law of Cosines' — they're the same formula, with the right triangle being the simplest case.
Heron's formula gives you area from three sides alone — no angles needed. Useful when you've got SSS data and just want area without finding any angles. A = √(s(s−a)(s−b)(s−c)) where s is the semi-perimeter. The calculator uses an equivalent formula internally, but Heron's is what gets cited in geometry textbooks.
Honest take: surveyors, civil engineers, and roofers actually use this math every day. Triangulating distances from two known points, computing roof pitches, laying out angles for foundations — all SSA, SAS, ASA work. If you're in construction or surveying, knowing your triangle cases isn't academic; it's a job skill.

?What is the Triangle Solver?

The Formula

Law of cosines: c² = a² + b² − 2ab·cos(C). Law of sines: a/sin(A) = b/sin(B) = c/sin(C). Area = ½·ab·sin(C). Pythagoras (right triangle): c² = a² + b².

Practical Examples

3-4-5 right triangle: SSS check — area 6, angles 36.87° / 53.13° / 90°.

Surveying: from two known angles and a baseline distance (ASA), compute the distance to a far point.

Carpentry: cutting a roof rafter — given run + rise, find the hypotenuse and angle.

Navigation: given two bearings to a landmark and the distance between observation points (AAS), triangulate the landmark's position.

Equilateral triangle (60-60-60): every triangle with all three angles equal is also equilateral (all sides equal).

Sum of any two sides MUST exceed the third — the triangle inequality. The calculator catches violations.

Frequently Asked Questions

Popular Conversions

Jump to a ready-made conversion — useful for quick reference and sharing:

Browse all Math

GPA Calculator

Calculate your semester or cumulative Grade Point Average on the 4.0 scale, weighted by credit hours, with support for both weighted and unweighted grading systems.

Grade Calculator

Calculate your final grade from weighted categories like tests, homework, and exams.

Student Result Card Calculator

Subject-wise marks, grades, and a printable school result card — ideal for small schools and parents.

Fraction Calculator

Add, subtract, multiply, and divide proper, improper, and mixed fractions with automatic reduction to lowest terms and decimal equivalents for cooking, carpentry, and homework.

Ratio Calculator

Simplify ratios to lowest terms using GCD and solve proportions like a:b = c:x for any missing value — useful for recipes, map scales, aspect ratios, and mixing ingredients.

Average Calculator

Compute the arithmetic mean, median, mode, range, count, sum, minimum, and maximum from any list of numbers — essential for descriptive statistics, surveys, and grading.

LCM / GCD Calculator

Find the LCM and GCD (HCF) of two or more integers using the Euclidean algorithm — useful for simplifying fractions, finding common denominators, and scheduling repeating events.

Prime Number Checker

Test whether any integer is a prime number using trial division up to its square root, and view the complete unique prime factorization for composite numbers.

Browse all Math calculators & converters

See every tool in this category, plus FAQs and category-specific guides.

Browse other categories

Islamic9 Financial14 Gold & Silver8 Health25 Unit19 Date & Time9 Construction22 Utility20 Fun12

Triangle Solver — Solve any triangle from SSS | Premium Converters

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Triangle Solver

Triangle Solver

Triangle Solver — SSS

Triangle (drawn to scale)

All sides & angles

Step-by-step calculation

Overview — Triangle Solver

From the editors — what to know before you start

Pro tips most people learn the hard way

?What is the Triangle Solver?

The Formula

Practical Examples

Frequently Asked Questions

Popular Conversions

Embed this calculator on your site

Browse other categories

Premium Converters

Triangle Solver

Triangle Solver

Triangle Solver — SSS

Triangle (drawn to scale)

All sides & angles

Step-by-step calculation

Overview — Triangle Solver

From the editors — what to know before you start

Pro tips most people learn the hard way

?What is the Triangle Solver?

The Formula

Practical Examples

Frequently Asked Questions

Popular Conversions

Embed this calculator on your site

Browse other categories

Premium Converters

Premium Converters

Triangle Solver

Triangle Solver

Triangle (drawn to scale)

All sides & angles

Step-by-step calculation

Overview — Triangle Solver

From the editors — what to know before you start

Pro tips most people learn the hard way

?What is the Triangle Solver?

The Formula

Practical Examples

Frequently Asked Questions

What's the SSA ambiguous case?

What's the difference between SAS and SSA?

Why does my SSS input fail?

How is area computed?

What units does the calculator use?

Popular Conversions

Embed this calculator on your site

More Math Tools

Browse other categories

Premium Converters

Triangle Solver

Triangle Solver

Triangle (drawn to scale)

All sides & angles

Step-by-step calculation

Overview — Triangle Solver

From the editors — what to know before you start

Pro tips most people learn the hard way

?What is the Triangle Solver?

The Formula

Practical Examples

Frequently Asked Questions

What's the SSA ambiguous case?

What's the difference between SAS and SSA?

Why does my SSS input fail?

How is area computed?

What units does the calculator use?

Popular Conversions

Embed this calculator on your site

More Math Tools

Browse other categories