How this instrument works
Side-angle-side is one of the classical triangle congruence conditions taught alongside SSS, ASA, and AAS: fix two sides and the angle squeezed between them, and exactly one triangle satisfies those three numbers, with nothing left undetermined. This sheet carries that certainty all the way through rather than stopping at a single figure — it runs the Law of Cosines to close the third side, then the Law of Sines to recover the two angles that were never given, and finally the plain area rule, so one side-angle-side reading turns into a complete triangle: three sides, three angles, one area.
The four results are not independent of one another; they come out of a strict chain. Side c (opposite C) is found first, straight from c² = a² + b² − 2ab·cosC, because nothing else can be pinned down until the triangle's size is fixed. Only then does Angle A follow, from the ratio a ⁄ sinA = c ⁄ sinC, since that Law of Sines equation needs c already in hand to leave exactly one unknown. Angle B costs nothing further — 180° − C − A finishes it by plain subtraction — and Area, ½ab·sinC, never needed either angle at all, since it depends only on the two original sides and the angle already known between them.
Watch what happens as the included angle C is pushed toward its two extremes with a and b held fixed. Near C = 0°, sides a and b fold back on themselves, Side c collapses toward |a − b|, Area drops toward zero, Angle A shrinks toward 0°, and Angle B opens toward a nearly flat 180°. Near C = 180°, the two sides straighten into an almost-straight line, Side c climbs toward a + b, Area again falls to zero, and this time both remaining angles collapse toward 0° together. Four numbers drifting toward their limits in step is the geometry itself giving out, not a rounding artifact.
- Enter the two known side lengths into Side a and Side b, in any consistent unit — every length in the results shares that same unit.
- Enter the angle trapped between those two sides into Included angle C, choosing degrees, radians, or turns from its unit menu.
- Double-check that C is the angle where sides a and b actually meet, not one of the triangle's other two corners — the whole chain depends on that being the included angle.
- Read Side c (opposite C) for the third length, then Angle A and Angle B for the two angles the sheet recovers.
- Read Area for the enclosed region — it comes out squared in whatever unit you used for the sides.
Worked example — 5 and 7 enclosing 60°
Take Side a = 5, Side b = 7, and Included angle C = 60°, which is 1.0471975511965976 radians once converted. The Law of Cosines closes the triangle first: c² = 5² + 7² − 2×5×7×cos(60°) = 74 − 35 = 39, so Side c (opposite C) = √39 = 6.244997998398397 — the exact figure this sheet returns before either remaining angle is touched.
Angle A comes from the Law of Sines now that c is known: sinA = 5×sin(60°) ⁄ 6.244997998398397 ≈ 0.693375, so Angle A = asin(0.693375) = 0.7661626496937841 radians, about 43.90°. Angle B needs no trig step of its own: 180° − 60° − 43.90° ≈ 76.10°, or 1.3282324526994116 radians, the exact value this sheet displays.
Area follows straight from the original two sides and the included angle, untouched by either derived angle: Area = ½ × 5 × 7 × sin(60°) = 17.5 × 0.8660254037844386 = 15.155444566227676 square units. A fencer reading a plot this way — two runs of 5 m and 7 m meeting at a 60° corner post — gets the missing boundary length, both remaining corner angles, and the enclosed ground, all from one pair of sides and the angle between them.
Questions
What does side-angle-side determine about a triangle?
Everything. Fixing two sides and the angle trapped between them pins down one and only one triangle — the Law of Cosines finds the third side, the Law of Sines then finds the two remaining angles, and ½ab·sinC finds the area, with no second possibility at any step. That is unlike the ambiguous side-side-angle case, where a non-included angle can leave zero, one, or two valid triangles for the same three numbers.
How is this different from the cosine-triangle third-side calculator?
That sheet stops at the Law of Cosines and reports only Side c (opposite C). This one keeps going from the same side-angle-side input: it takes the c that formula produces, feeds it into the Law of Sines to recover Angle A and Angle B, and adds the area — the complete solve rather than the single length.
Why is Angle B found by subtraction instead of another Law of Sines step?
Because a triangle's three angles always sum to 180°, so once C is given and Angle A is recovered from the Law of Sines, Angle B = 180° − C − A costs nothing further. Skipping a second inverse-sine calculation also sidesteps any risk of picking the wrong branch, since plain subtraction cannot be ambiguous the way an arcsine step can.
Does the area formula need either of the two computed angles?
No — Area = ½ab·sinC uses only the two original sides and the included angle between them, the same three numbers used to find the third side. That is why the area on this sheet matches a standalone SAS-area calculator exactly for the same a, b, and C, even though this sheet also reports Side c, Angle A, and Angle B alongside it.
What happens if I enter the angle at the wrong corner?
The formulas still return numbers, but not the right ones — every step here assumes C is the angle physically enclosed by sides a and b, not one of the triangle's other two corners. Swap in a non-included angle and Side c, Angle A, Angle B, and Area all come out wrong together, since each one depends on that same misidentified C.
Can side-angle-side ever fail to produce a valid triangle?
Not for any positive a, b and any C strictly between 0° and 180° — that combination always closes into a real triangle, with Side c landing strictly between |a − b| and a + b. Only at the two excluded extremes, where C flattens to 0° or 180°, does the shape collapse into a line and Area fall to zero.