Triangle Calculator (right angle, SSS, SAS)
Pick a mode, enter known values, and get all sides, all three angles, area, and perimeter. Validates triangle inequality for SSS mode.
Sides
- a
- 3
- b
- 4
- c
- 5
Angles
- A
- 36.8699°
- B
- 53.1301°
- C
- 90°
How it works
Three modes for three problems
Right triangle (legs a, b): the most common case. Hypotenuse c = √(a² + b²) by the Pythagorean theorem. Angle A = arctan(a/b), B = arctan(b/a), C = 90°. Area = (a × b) / 2.
Three sides (SSS — Side-Side-Side): when you know all three side lengths. Validate with triangle inequality (longest side < sum of other two). Angles via law of cosines: cos A = (b² + c² − a²) / (2bc). Area via Heron's formula: √(s(s−a)(s−b)(s−c)) where s = (a+b+c)/2.
Two sides + included angle (SAS — Side-Angle-Side): when you know two sides and the angle between them. Third side via law of cosines: c = √(a² + b² − 2ab cos C). Other angles via law of sines. Area = (1/2) × a × b × sin C.
When to use each mode
Right triangle is the most common in everyday geometry — building a wall corner, calculating a roof rise, finding a TV's diagonal from width and height. The Pythagorean theorem is one of the most-used formulas in construction and design.
SSS shows up in surveying, navigation, and any case where you measure three sides directly. Useful for verifying a triangle's shape from physical measurement.
SAS is for cases where you have two sides meeting at a known angle but the third side is unmeasured or hard to reach. Common in trigonometry classes and trigonometric problem sets.
Practical applications
Construction: roof pitch from rise and run (right triangle). Carpenter's 'rule of three': a 3-4-5 triangle has a perfect right angle, no protractor needed.
Navigation: triangulation uses SSS or SAS to find your position from three known landmarks. The same math powers GPS (with relativistic corrections).
Computer graphics: every 3D model is decomposed into triangles. The triangle area formula appears in shader code computing surface lighting.
Astronomy: parallax distance measurements use SSS principles. The sun-Earth-star angle gives the star's distance via simple trig.
Frequently asked questions
›What's the triangle inequality?
For three lengths to form a triangle, the sum of any two must exceed the third. So 3, 4, 5 is valid (3+4=7 > 5); 1, 2, 5 is not (1+2=3 < 5). The calculator catches this in SSS mode.
›How accurate is this?
Floating-point precision (~15-17 significant digits). For typical inputs the error is much smaller than measurement error in the real world.
›Can angles be in radians?
Not yet — we use degrees throughout. Convert via radians × 180/π if needed, or use our angle converter.
›What if my triangle has obtuse angle?
All modes handle obtuse triangles correctly. Make sure for SAS that the included angle is between 0° and 180° exclusive.
›Why doesn't this work for all 'two sides and an angle'?
We support SAS (the angle BETWEEN two sides). The ambiguous case (SSA — angle, then two sides where one is opposite the angle) sometimes has 0, 1, or 2 valid triangles. SSS and SAS are unambiguous.
›Can I solve obtuse 3-4-5 type problems?
3-4-5 is a right triangle. For obtuse, try 4-5-7 in SSS mode — the largest angle is over 90°.
›How do I find side from angles only?
You can't — angles alone determine shape but not size. You need at least one side. Two angles + one side is enough (third angle = 180 − sum of two).
›Does the data leave my browser?
No. Calculation runs locally; nothing is sent to a server.
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