Triangle Calculator (SSS / SAS / ASA)
Solve any triangle from three sides (SSS), two sides plus the included angle (SAS), or two angles plus the included side (ASA). Instantly returns the remaining sides and angles plus area, perimeter and classification (equilateral / isosceles / scalene, right / acute / obtuse).
Three side lengths — angles are recovered via the law of cosines.
Sides must satisfy the triangle inequality — the sum of any two sides must exceed the third.
Sides
Side a
3
Given
Side b
4
Given
Side c
5
Given
Interior angles
∠A (opposite a)
36.87°
Derived
∠B (opposite b)
53.13°
Derived
∠C (opposite c)
90.00°
Derived
Area
6
Perimeter
12
Classification
Scalene · Right
Unitless — enter all sides in the same unit (cm, m, inches, feet) and results are reported in that same unit. Angles are always shown in degrees (°).
Formula
SSS: cos A = (b² + c² − a²) / (2 b c) | SAS: a = √(b² + c² − 2 b c · cos A) | ASA: a / sin A = b / sin B = c / sin C | Area = ½ · b · c · sin A
- · Naming convention: side a is opposite ∠A, side b opposite ∠B, side c opposite ∠C. The three interior angles always sum to 180°.
- · SSS mode: the three sides must satisfy the triangle inequality — the sum of any two sides must strictly exceed the third, otherwise no triangle exists.
- · SAS mode: the entered angle ∠A is the angle between the two known sides b and c; that placement is what makes the solution unique.
- · ASA mode: the entered side c is the side between the two known angles ∠A and ∠B; the two angles must sum to less than 180°.
- · SSA (two sides + a non-included angle) is intentionally not supported — it is the classical ambiguous case and can yield 0, 1 or 2 valid triangles depending on the inputs.
- · Sides are unitless: enter all three in the same unit (cm, m, in, ft) and the result uses that unit. Angles are always reported in degrees.
- · References: Stewart, "Calculus: Early Transcendentals" 8th ed., Appendix D; Bronshtein & Semendyayev, "Handbook of Mathematics" 6th ed., §3.2.
Frequently asked
Why does the calculator only accept SSS, SAS and ASA inputs?
SSS, SAS and ASA each pin down exactly one triangle, so the law of cosines (SSS / SAS) and the law of sines (ASA) give a single answer with no ambiguity. AAS — two angles plus a non-included side — is just ASA in disguise: subtract the two known angles from 180° to recover the third, and it becomes ASA. SSA (two sides plus a non-included angle) is the famous ambiguous case: the same inputs can correspond to 0, 1 or 2 valid triangles, so we intentionally exclude it rather than silently picking one.
What unit should I use for the sides — does inches or feet work?
The calculator is unit-agnostic — you can enter cm, metres, inches, feet, pixels, even light-years, as long as all sides use the same unit. The returned sides and perimeter come out in that same unit, and the area is in the corresponding squared unit (e.g. metres in → m² out; feet in → ft² out). Angles are always shown in degrees regardless of which length unit you used.
I entered three sides and got a "triangle inequality" error — what does that mean?
The triangle inequality says that the sum of any two sides must strictly exceed the third side. For example 2, 3, 5 fails because 2 + 3 = 5 — the three lengths collapse onto a straight line with zero area instead of enclosing a triangle. 2, 3, 6 is even further from being valid. Check whether you have a typo, especially that the longest side is not greater than or equal to the sum of the other two.
How is the area computed — does it match Heron's formula?
It matches Heron's formula to within floating-point error. The widget uses ½ · b · c · sin A directly because, in every supported mode, ∠A and the two adjacent sides b, c are known (in SSS mode, ∠A is recovered first via the law of cosines). Heron's formula √(s(s-a)(s-b)(s-c)), where s is the semi-perimeter, is mathematically equivalent. We prefer ½ b c sin A because it stays better-conditioned numerically for very thin, near-degenerate triangles.
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