Triangle Length Calculator - SAS Law of Cosines

The triangle length calculator finds the missing third side of any triangle when two side lengths and the included angle are known. It applies the law of cosines, converts the included angle from degrees to radians, and returns the closing side in the same length unit as the inputs.

• • Geometry homework: Drop in the two given side lengths and the angle between them, then copy the third side into the next step.
• • Roof and rafter layout: Translate a measured wall plate, a rafter length, and the peak angle into the rafter rise or the diagonal run.
• • Land and survey plots: Compute a missing boundary distance from two tape-measured sides and the angle at the shared corner.
• • Navigation and bearing problems: Estimate the closing leg from two course distances and the bearing change at the turn.

The law of cosines, also called the cosine rule, is the right tool for the SAS case. The Pythagorean theorem is the same formula with the included angle fixed at 90 degrees, so the calculator handles right triangles as a built-in case rather than a separate mode.

Keep every input in the same length system. The angle is read in degrees no matter the length unit, and the returned side is expressed in whatever length unit was used for the inputs.

When the triangle being measured is clearly a right triangle, the Right Triangle Calculator can solve the hypotenuse directly without a separate law of cosines step.

The triangle length calculator takes the two side lengths and the included angle, converts the angle to radians, and returns the third side as the square root of a squared plus b squared minus two times a times b times the cosine of the included angle. The cosine of the included angle and the triangle perimeter are shown alongside the third side so each step stays auditable.

• side a: first known length of the triangle, in any length unit
• side b: second known length of the triangle, sharing the included angle with side a
• included angle gamma: angle between sides a and b, entered in degrees and converted to radians for the cosine
• third side c: computed length of the side opposite the included angle, returned in the same length unit as the inputs

The cosine of the included angle carries the geometry. When the angle is acute, the cosine is positive and the third side is shorter than the simple sum of the two inputs. When the angle is obtuse, the cosine is negative and the third side grows longer.

Converting the angle to radians is a small step with real consequences. Skipping it would evaluate the cosine of a degree value and return a meaningless third side, so the calculator performs the conversion before any trigonometry.

According to Wolfram MathWorld, the law of cosines states that c^2 = a^2 + b^2 - 2ab cos(gamma), and reduces to the Pythagorean theorem when gamma equals 90 degrees.

Once the third side is known, the Triangle Area Calculator turns the same side lengths into the area of the triangle using base-height, Heron's formula, or the SAS area rule.

These four ideas decide whether the law of cosines is the right tool and how to read the result.

When the job is to recover all three sides and all three angles at once, the Triangle Calculator accepts the same SAS measurements and returns the full triangle solution.

Read the two side lengths and the angle between them from your problem, drop them into the form, and use the third side as soon as it appears.

1. 1 Measure the two known sides: Pick the two sides of the triangle that share a common vertex, then measure each in the same length unit.
2. 2 Read the included angle: The included angle sits at the vertex where the two measured sides meet. Read it in degrees.
3. 3 Enter the values: Type the first side into side a, the second into side b, and the included angle into the angle field. The result updates as soon as the third value is in place.
4. 4 Read the third side: The primary result is side c, returned in the same length unit as the inputs.
5. 5 Audit the cosine and the perimeter: The cosine of the included angle and the triangle perimeter keep each step auditable.
6. 6 Fix invalid input errors: Check for a zero or negative side, a missing field, or an included angle outside (0, 180) degrees.

A roof gable has a wall plate of 24 feet, a rafter length of 16 feet, and a peak angle of 50 degrees at the plate. Enter side a = 24, side b = 16, and included angle = 50. The calculator returns c = sqrt(24^2 + 16^2 - 2 * 24 * 16 * cos(50 deg)) = 19.15 length units, which is the diagonal run of the rafter across the plate.

After the third side is known, the Perimeter of a Triangle with Fractions Calculator adds the three side lengths into a triangle perimeter that handles fractional measurements cleanly.

Pulling the law of cosines into a single form keeps the missing side one click away from the SAS measurements.

• • Direct SAS support: Two sides and the included angle are exactly the inputs the law of cosines needs, so the third side comes back without intermediate steps.
• • Real-time updates: Every change to a side length or the included angle updates the third side on the spot.
• • Auditable support values: The cosine of the included angle and the triangle perimeter appear next to the third side.
• • Built-in Pythagorean case: An included angle of 90 degrees reduces the law of cosines to the Pythagorean theorem, so the calculator handles right triangles without a separate mode.
• • Unit-agnostic length output: The same calculator handles inches, feet, centimeters, and meters because the third-side label follows the input length unit.
• • Input validation: Zero or negative sides, missing fields, and angles outside (0, 180) degrees are caught and explained instead of returning a misleading zero or a NaN.

The SAS input pattern is the natural choice for two sides and the angle between them. Right triangles are the boundary case. Setting the included angle to 90 degrees turns the law of cosines into the Pythagorean theorem, which keeps the calculator useful for the most familiar triangle type without a separate mode.

If the larger shape is a polygon that has to be split into triangles before its area can be found, the Polygon Area Calculator uses the same side-and-angle measurements to set up each piece.

The math is exact, but the precision of the inputs and the unit choice decide how trustworthy the third side is.

• • The law of cosines assumes the included angle sits between the two known sides. An angle to a different vertex produces a wrong closing side.
• • The cosine function in the browser rounds near 0 and 180 degrees, so an angle reported as exactly 180 would return a slightly negative radicand. The calculator clamps that to zero but also rejects the input.
• • The calculator returns a geometric length only. Real measurements may need to add tolerance for tape stretch, plumb bob sag, or surface roughness.

If the third side seems off, the most common cause is that the entered angle is not the included angle. Check that the angle is read at the vertex shared by the two known sides.

According to Khan Academy, the law of cosines generalizes the Pythagorean theorem by relating the three sides of any triangle to one of its angles, with c^2 = a^2 + b^2 - 2ab cos(C).

According to OpenStax, the law of cosines can be used to find the length of a missing side of any triangle when two sides and the included angle are known.

When the layout also includes a rectangular piece that needs its own quick length times width estimate, the Length Width Area Rectangle Calculator keeps the simpler shape on the same workflow.