Free · Triangle calculators 🔺

Triangle Sides and Area from Included Side and End Angles

Side a joins the vertices carrying β and γ. It lies opposite the computed angle α. The side lengths follow the sine law, and the area uses the included-angle triangle formula.

Formula & worked example
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How to use this calculator

Enter the known values in the units shown. Results update as you type.
Where results are editable, change one to solve backwards. Lock a value to hold it fixed.
Use the worked example to check the method. Reset restores the starting fields.

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Formula and method

Example in initial displayed units: a 4, Angle β 30, Angle γ 60 gives Angle α 90, b 2, c 3.464102, Area 3.464102.

α = π−β−γ; b/a = sin β/sin α; area = ab sin γ/2

Worked example

Enter these known values and leave the other values blank.

a: 4 cm
Angle β: 30 deg
Angle γ: 60 deg

Angle α: 90 deg
b: 2 cm
c: 3.464 cm
Area: 3.464 cm²

Assumptions and limitations

Side a lies between β and γ and opposite α. Enter the included side and its two endpoint angles. Other side lengths, the remaining angle and area are one-way, read-only results.
Use compatible units and enough independent values. Clear unknowns before solving backward. Original reference drawings are schematic and do not scale to entered values.

Common questions

Which side is between the two entered angles?