Triangle Side Angle Calculator - One Side and Two Angles

A triangle side angle calculator solves a triangle when you know one side and the two angles that share the unknown sides. You enter the known side a and the two known angles B and C, and the tool returns the missing third angle A, the two missing sides b and c, the area, and the perimeter.

• • Roof and rafter layout: Enter the rafter length and the two pitch angles to recover the run, rise, and roof area.
• • Surveying a plot: Measure one leg and read the two endpoint angles to recover the other sides and area.
• • Navigation legs: Use a known distance and two heading changes to back out the new leg lengths.
• • Cross-checking homework: Confirm the law of sines step returns the same answer.

The triangle side angle calculator is the AAS (Angle-Angle-Side) case in plain language. The two given angles and the single side determine the triangle uniquely, so the third angle, the two missing sides, the area, and the perimeter all come out of a short fixed sequence of formulas.

The side-and-angle input pattern is the same one the AAS triangle calculator uses, so swapping between the two is a matter of which side you measure first, not which formulas you run.

The calculator first uses the angle sum of a Euclidean triangle to recover the third angle from the two known angles, then uses the law of sines to scale the known side to the two missing sides, and finishes with the half-base-times-height area identity and a perimeter sum.

• a: The single known side, opposite the unknown angle A. The other two sides scale to a through the law of sines.
• B, C: The two known angles, in degrees, opposite the sides b and c that the calculator returns. B and C must each be in (0, 180) and must sum to strictly less than 180.
• A: The missing third angle, recovered from the angle sum 180 - B - C, then used as the denominator in the law of sines.
• b, c: The two missing sides, recovered from a * sin(B) / sin(A) and a * sin(C) / sin(A) respectively.
• Area, P: Area from 0.5 * a * b * sin(C) and perimeter from a + b + c.

The law of sines is a proportion that holds for every Euclidean triangle, and the angle sum identity closes the loop when only one angle is missing. Together they turn the AAS case into a short sequence of mechanical steps.

According to Wikipedia, Law of sines, the law of sines states that the ratios of the side lengths of a triangle to the sines of the opposite angles are equal as a/sin(A) = b/sin(B) = c/sin(C), and together with the angle sum A + B + C = 180 degrees it uniquely solves a triangle when one side and the two adjacent angles (the AAS case) are known

When you want to focus only on the law of sines identity itself, the law of sines calculator accepts a single side-angle pair and a target angle or side and runs the same ratio without the angle-sum step.

These four ideas cover what makes the side-and-angle case different from SSS, SAS, or ASA.

These four pieces turn the side-and-angle problem into a short sequence of mechanical steps. The angle sum gives A, the law of sines scales a to b and c, and the half-base-times-height formula gives the area.

If your two given quantities are both angles and the included side is known, the ASA triangle calculator is the natural sibling and returns the third angle from the angle sum before the law of sines runs.

The result panel updates as you type, so you can treat the form as a scratch pad for the AAS case.

1. 1 Enter side a: Type the length of the single known side. Use a positive number and pick a unit that matches the rest of the problem (cm, mm, m, in, or ft).
2. 2 Enter angle B: Type the first known angle in degrees, strictly between 0 and 180. This is the angle opposite the side b the calculator will return.
3. 3 Enter angle C: Type the second known angle in degrees. B and C must sum to strictly less than 180, otherwise the third angle A would not be positive.
4. 4 Pick a length unit label: Choose the length unit (cm, mm, m, in, or ft) you want shown next to side b, side c, the perimeter, and the area. The label changes only the displayed text, not the calculation.
5. 5 Read the third angle A: Confirm that A is positive. If A is zero or negative, the two given angles already sum to 180 or more and the form shows a validation error.
6. 6 Read sides b and c, the area, and the perimeter: Use b and c for the triangle's other two sides, the area for surface or material estimates, and the perimeter for trim, fencing, or framing totals.

For a right-rafter layout with a 4 m rafter on a 30 deg wall plate meeting the ridge at 60 deg, enter a = 4, B = 30, C = 60. The calculator returns A = 90 deg, b = 2.00 m, c = 3.46 m, area = 3.46 sq m, perimeter = 9.46 m. Use b for the run and c for the rise.

For problems where you know all three side lengths and need to back out the angles, triangle length calculator applies the law of cosines in reverse and gives you the same final side set from a different starting point.

Working through the AAS case by hand is fine for one problem, but the calculator removes a few classes of error that show up when you have to do it repeatedly.

• • One pass, six outputs: Angle A, sides b and c, area, perimeter, and unit labels come out of one input.
• • Avoids the SSA ambiguity: One side and two angles uniquely define the configuration.
• • Catches impossible inputs early: If B or C is outside (0, 180), the form flags an error.
• • Real-time updates for scratch work: The result panel refreshes on every keystroke, so you can sweep B and watch the triangle update.
• • Unit-agnostic by design: Sides, perimeter, and area return in the unit you entered.

These benefits matter most when the input data comes from a real measurement rather than a textbook problem.

Once the calculator has returned all three sides, the triangle area calculator cross-checks the area using Heron's formula so you can confirm the answer with an independent identity.

Four things change the numbers the calculator returns, and the limitations below describe what the law of sines does and does not cover.

• • The law of sines only applies to planar Euclidean triangles. It does not solve spherical triangles (for example, great-circle navigation on a globe) without the spherical version of the identity.
• • When the two given angles already sum to 180 degrees or more, the third angle A would be zero or negative and the denominator sin(A) would be zero. The calculator flags the input as invalid before the law of sines is applied.

These factors are why the calculator asks you to keep each angle strictly between 0 and 180 degrees, and to keep the sum B + C strictly below 180.

According to Omni Calculator, Triangle side angle, knowing one side and the two angles that share the unknown sides is enough to uniquely define a triangle, and the missing angles, missing sides, area, and perimeter can be computed with the angle sum, the law of sines, and the half-base-times-height area identity

If your angles are recorded in gradians, radians, or turns instead of degrees, the angle converter converts them to degrees before you enter them into the triangle side angle calculator.