Triangle Calculator (SAS) — Two Sides & Included Angle

Last updated: 2026-05-18

Solving a Triangle from Two Sides and the Included Angle

SAS (Side-Angle-Side) is the triangle configuration in which two side lengths and the angle enclosed between them are known. From these three quantities the rest of the triangle is fully determined: the third side, the two remaining angles, the area, and the inscribed and circumscribed circle radii. The Law of Cosines supplies the missing side first, and a second application of it recovers the remaining angles.

Finding the Third Side

If sides $a$ and $b$ are known and $C$ is the angle between them, the third side $c$ (opposite $C$) is:

$c = \sqrt{a^2 + b^2 - 2ab\cos C}$

Worked example — $a = 5$, $b = 7$, $C = 60°$:

$c = \sqrt{25 + 49 - 2 \times 5 \times 7 \times \cos 60°} = \sqrt{74 - 35} = \sqrt{39} \approx 6.245$

When $C = 90°$, $\cos 90° = 0$ and the formula reduces to $c = \sqrt{a^2 + b^2}$ — the Pythagorean theorem.

Area from SAS

The area of the triangle follows directly from the two sides and included angle, without needing to find $c$ first:

$S = \frac{1}{2}\,ab\sin C$

For the example above: $S = \tfrac{1}{2} \times 5 \times 7 \times \sin 60° = \tfrac{35\sqrt{3}}{4} \approx 15.155$.

Recovering the Remaining Angles

Once $c$ is known, treat the triangle as SSS and apply the Law of Cosines again to find $A$:

$A = \arccos\!\left(\frac{b^2 + c^2 - a^2}{2bc}\right)$

Then $B = 180° - A - C$.

Alternatively, the Law of Sines ($a/\sin A = c/\sin C$) gives $A = \arcsin(a\sin C / c)$, but the cosine form is preferred because it avoids the ambiguous case that arises with arcsin when the angle might be obtuse.

Circumradius and Inradius

With all three sides and the area known:

$R = \frac{abc}{4S}, \qquad r = \frac{S}{s} \quad\text{where } s = \frac{a+b+c}{2}$

For the 3-4-5 right-triangle SAS case ($a=3$, $b=4$, $C=90°$): $c=5$, $S=6$, $R=2.5$, $r=1$.

SAS vs. SSS

SAS applies when the included angle is measured directly (for example with a protractor or surveying instrument). Once the third side is computed, the problem reduces to SSS.

Distinction from the ambiguous case (SSA)

SAS requires the angle to lie between the two known sides. When the known angle is opposite one of the known sides instead, the configuration is SSA (Side-Side-Angle), which can produce zero, one, or two valid triangles. This calculator handles only the unambiguous SAS case.