How this instrument works
SAS stands for side-angle-side: you know two sides of a triangle, call them a and b, and the angle C squeezed between them. That is already enough information to fix the triangle's shape and size, so its area follows without needing the third side or the height. The formula drops a perpendicular from the vertex where a and b meet down onto side a, and that perpendicular has length b·sin(C) — trigonometry supplying the height that a tape measure would otherwise have to find. Multiply by a and halve it, and A = ½ab·sin(C) falls straight out of the ordinary base-times-height rule.
A property worth sitting with: for two side lengths held fixed, area rises and falls with sin(C) alone, and sine peaks at exactly one right angle. Swing the angle between a and b from a sliver up toward a straight line and the enclosed area climbs, tops out the instant C reaches 90°, then falls away again as the triangle flattens toward the far side. Among every triangle you could build from a given pair of sides, the roomiest one is always the one with a right angle pinned between them — a fact that has nothing to do with either side's length and everything to do with the shape of sine.
The angle has to be the included one — the one actually sitting between the two sides you measured, not some other corner of the triangle. Feed in a non-included angle and the formula returns a number, but not the triangle's area; there is no shortcut around identifying which corner the two known sides actually meet at. At the far edges of the domain the formula still behaves: as C shrinks toward 0° or grows toward 180°, sin(C) shrinks toward zero and the triangle collapses into a straight line of zero area, exactly as a flattening wedge should.
- Enter the two known side lengths into Side a and Side b, in any consistent unit — area comes out squared in whatever unit you used.
- Enter the angle between those same two sides into Included angle C, choosing degrees, radians, or turns from its unit menu.
- Double-check that C is the angle actually enclosed by sides a and b, not one of the triangle's other two corners.
- Read the result straight off the Area field.
Worked example — a square corner between 5 and 7
A fenced plot has two straight boundary runs meeting at a corner post: one measured at 5 m (side a), the other at 7 m (side b), and a total station confirms the corner itself is a right angle, C = 90°. The area follows at once: A = ½ × 5 × 7 × sin(90°) = ½ × 35 × 1 = 17.5 square metres — the fencer's material order without ever chaining a diagonal or a height.
That clean result is the right-angle case doing its own quiet arithmetic: sin(90°) equals 1, so the SAS formula collapses exactly to the plain ½ × base × height rule, because side b is itself standing perpendicular to side a. Pivot that same corner post to 60° or to 120° while leaving both 5 m and 7 m runs untouched, and the enclosed area drops below 17.5 either way — 90° is not an arbitrary example here, it is the one angle that makes this particular pair of sides enclose the most ground possible.
Questions
Which angle counts as the included angle?
The one actually sitting between the two sides you measured — where a and b physically meet at a shared vertex. It is not the angle opposite either side, and not one of the triangle's other two corners. Plugging a non-included angle into A = ½ab·sin(C) produces a number, but not the correct area, because the formula's derivation depends on that specific perpendicular falling from that specific vertex.
Why does the SAS area formula use sine instead of cosine?
Sine measures the fraction of side b that stands perpendicular to side a, which is exactly the height that base-times-height area needs: h = b·sin(C). Cosine measures the fraction lying along side a instead, which is what the Law of Cosines uses to find the length of the third side, c² = a² + b² − 2ab·cos(C) — a related but different question from area.
What included angle gives the biggest triangle for two fixed sides?
Exactly 90°, since sin(C) reaches its maximum value of 1 there and nowhere else in a triangle's valid range. With sides of 5 and 7 held fixed, a 90° included angle gives the maximum possible area of 17.5; tilt that same corner to 30° instead and the area falls to ½ × 5 × 7 × sin(30°) = 8.75, even though neither side length changed.
Can two different triangles share the same side-angle-side data?
No — SAS pins a triangle down uniquely. Once a, b, and the included angle C are fixed, the Law of Cosines fixes the third side, and the Law of Sines then fixes the remaining two angles, leaving exactly one triangle. That certainty is why SAS, unlike side-side-angle, never produces the ambiguous two-triangle case that trips up Law of Sines problems.
Does the formula still work once the included angle passes 90 degrees?
Yes, across the whole valid range from just above 0° to just below 180°. Sine stays positive throughout that interval — sin(150°) is 0.5, the same value as sin(30°) — so an obtuse included angle still returns a positive, correct area. Only at the two extremes, where the triangle flattens into a straight line, does the area fall to zero.