SAS Triangle Calculator: Find a Missing Side or Angle

Yes—SAS triangle calculation can be fun because two sides and their included angle are enough to determine one unique triangle. Enter the two side lengths and the angle between them to calculate the third side and the two remaining angles.

Formula

SAS means side-angle-side: the known angle lies between the two known sides. The third side follows from the law of cosines, and the remaining angles can then be found with the same law and the triangle angle sum.

• Third side: c² = a² + b² − 2ab cos(C), so c = √(a² + b² − 2ab cos(C)).
• Angle A: A = arccos((b² + c² − a²) / (2bc)).
• Angle B: B = 180° − C − A.

Here, a and b are the known side lengths, c is the unknown side length, and A, B, and C are the angles opposite sides a, b, and c respectively. Use the same length unit for all three sides; the angles are in degrees in this calculator. The symbol cos means cosine and arccos means inverse cosine.

Worked Example

Use the calculator’s default values: a = 5 units, b = 7 units, and included angle C = 60°. The units could be centimetres, inches, or another consistent measure; the numerical calculation is unchanged.

1. Apply the law of cosines: c = √(5² + 7² − 2 × 5 × 7 × cos 60°).
2. Since cos 60° = 0.5, c = √(25 + 49 − 35) = √39 ≈ 6.24 units.
3. Find A: A = arccos((7² + 6.24² − 5²) / (2 × 7 × 6.24)) ≈ 44.42°.
4. Find B from the angle sum: B = 180° − 60° − 44.42° = 75.58°.

Rounded to two decimal places, the results are c = 6.24 units, A = 44.42°, and B = 75.58°. The unrounded values are used internally when calculating B, so small differences can appear if you repeat the calculation using only the displayed rounded side.

Why SAS Determines One Triangle

Two side lengths alone can form many triangles because the angle between those sides can change. Once that included angle is specified, the two sides and their orientation fix the triangle’s shape and size, apart from flipping it over. This is why SAS is a triangle-congruence condition: two triangles with the same two sides and included angle have the same dimensions.

The included-angle detail matters. If the angle you know is not between the two known sides, the information is not SAS; it may instead be side-side-angle, which can have zero, one, or two valid triangles. A calculator using the SAS formula cannot resolve that different case just by relabelling the inputs.

When to Use the Calculation

Use SAS when measurements provide two lengths and the angle where those sides meet. For example, a surveyor could record two distances from a shared point and the angle between their directions, then calculate the separation between the endpoints. The same geometry appears in navigation, construction drawings, physics diagrams, and classroom problems.

For a physical measurement, be clear about what the angle represents: it must be the interior angle between the two sides, not an exterior turn or a direction measured from a separate reference line. If the given angle is an exterior angle, convert it to the interior angle before applying the formula.

Reading Results and Avoiding Mistakes

The third side is in the same unit as the entered sides, while the other outputs are angles in degrees. The calculator labels side units as “units” because you choose the measurement system. If a side is in metres and another is in feet, convert one before calculation; mixing units makes the squared terms in the formula incompatible.

• Check the angle range: an ordinary interior triangle angle must be greater than 0° and less than 180°.
• Keep the angle included: C must sit between sides a and b under the stated labels.
• Delay rounding: rounding an intermediate side or angle can shift later results by a small amount.
• Check the angle sum: A + B + C should equal 180°, allowing for display rounding.
• Use degree mode: the formula shown uses cosd, meaning cosine in degrees. A hand calculation in a calculator set to radians may give a wrong result.

For the defaults, c is about 6.24, which is longer than the difference between the known sides (2) and shorter than their sum (12). Those inequalities are useful quick checks: a valid triangle’s third side must be greater than the difference in the other side lengths and less than their sum.

Limits and Related Methods

The law of cosines works for any valid SAS triangle, including acute, right, and obtuse included angles. At C = 90°, cos(C) is zero, so the formula reduces to the Pythagorean theorem, c² = a² + b². That familiar rule is therefore a special case rather than a competing formula.

The calculation assumes the inputs describe an exact triangle. Real measurements have uncertainty, and the calculator does not estimate that uncertainty or decide whether a measuring instrument was accurate. A very small angle or a value close to 180° can also make the triangle narrow and sensitive: a small measurement error may noticeably affect the calculated angles.

Other information patterns call for other methods. SSS (three sides) can be solved with the law of cosines; ASA or AAS (two angles and a side) is commonly handled with the law of sines; and right triangles can often use basic trigonometric ratios. Identify which values are known before choosing a formula.

Frequently Asked Questions

What does SAS mean in a triangle?
SAS stands for side-angle-side. It describes two known sides and the angle included between them.

Can I find a triangle with two sides and a non-included angle?
That is generally an SSA setup, not SAS. SSA can produce two different triangles, one triangle, or none, so it needs a separate case check.

Why use the law of cosines instead of the Pythagorean theorem?
The Pythagorean theorem applies directly only to right triangles. The law of cosines handles any included angle and becomes the Pythagorean theorem when that angle is 90°.

Do the side lengths need to be in centimetres or inches?
No particular unit is required, but both input sides must use the same unit. The calculated third side will use that same unit.

Why do my calculated angles add up to slightly less or more than 180°?
Displayed values are rounded. Check the unrounded calculations or allow a small rounding difference; the exact interior angles of a triangle sum to 180°.

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in a | Side a | units | 5 | 0
in b | Side b | units | 7 | 0
in C | Included angle C | degrees | 60 | 0
out c | Third side c | units | sqrt(a^2 + b^2 - 2*a*b*cosd(C)) | 2
out A | Angle A | degrees | acosd((b^2 + c^2 - a^2)/(2*b*c)) | 2
out B | Angle B | degrees | 180 - C - A | 2