Triangle Calculator (SSS & SAS)
Area, angles and missing side of any triangle.
Inputs
Use this calculator on your own site
Free embed
- Two lines of HTML, unlimited views
- Shows a small CalculateEveryday bar, with a credit link under the calculator
Formula
Law of Cosines: c² = a² + b² − 2ab·cos C. Heron's formula: Area = √(s(s − a)(s − b)(s − c)), with s = (a + b + c) ÷ 2
With three sides known, the Law of Cosines rearranged gives each angle and Heron's formula gives the area without needing a height. With two sides and the angle between them, the Law of Cosines gives the third side directly and the area is ½·a·b·sin C. Angle A is opposite side a, B opposite b, and C opposite c.
Worked example
For sides 3, 4 and 5 the semi-perimeter is 6, so the area is √(6 × 3 × 2 × 1) = 6. The Law of Cosines gives angles of 36.87°, 53.13° and exactly 90°: a right, scalene triangle with perimeter 12.
Where this can give the wrong answer
- Three lengths only make a triangle if the two shorter ones add up to more than the longest. 1, 2 and 3 lie flat in a line, and the page says so rather than returning an area of 0.
- Only one box is ignored at a time: side c is calculated in SAS mode, and angle C is calculated in SSS mode. Whatever is typed in the ignored box has no effect.
- Two sides and an angle that is not between them (SSA) can describe two different triangles, one, or none. That ambiguous case is deliberately not offered here.
- Angles are in degrees and shown to two decimals, so three rounded angles can add to 179.99° or 180.01°.
FAQ
- C is the angle between sides a and b, and it faces side c. The same pairing holds all round: each angle is named after the side opposite it.
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