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
- 36.8699ยฐ
- B
- 53.1301ยฐ
- C
- 90ยฐ
How it works
Three modes for three problems
Right triangle (legs a, b): the most common case. Hypotenuse c = โ(aยฒ + bยฒ) by the Pythagorean theorem. Angle A = arctan(a/b), B = arctan(b/a), C = 90ยฐ. Area = (a ร b) / 2.
Three sides (SSS โ 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 = (bยฒ + cยฒ โ aยฒ) / (2bc). Area via Heron's formula: โ(s(sโa)(sโb)(sโc)) where s = (a+b+c)/2.
Two sides + included angle (SAS โ Side-Angle-Side): when you know two sides and the angle between them. Third side via law of cosines: c = โ(aยฒ + bยฒ โ 2ab cos C). Other angles via law of sines. Area = (1/2) ร a ร b ร sin C.
When to use each mode
Right triangle is the most common in everyday geometry โ 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
โบWhat'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.
โบHow 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.
โบCan angles be in radians?
Not yet โ we use degrees throughout. Convert via radians ร 180/ฯ if needed, or use our angle converter.
โบWhat if my triangle has obtuse angle?
All modes handle obtuse triangles correctly. Make sure for SAS that the included angle is between 0ยฐ and 180ยฐ exclusive.
โบWhy doesn't this work for all 'two sides and an angle'?
We support SAS (the angle BETWEEN two sides). The ambiguous case (SSA โ angle, then two sides where one is opposite the angle) sometimes has 0, 1, or 2 valid triangles. SSS and SAS are unambiguous.
โบCan I solve obtuse 3-4-5 type problems?
3-4-5 is a right triangle. For obtuse, try 4-5-7 in SSS mode โ the largest angle is over 90ยฐ.
โบHow do I find side from angles only?
You can't โ angles alone determine shape but not size. You need at least one side. Two angles + one side is enough (third angle = 180 โ sum of two).
โบDoes the data leave my browser?
No. Calculation runs locally; nothing is sent to a server.
Related tools
Last updated: