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Triangle Calculator
Enter what you know about a triangle and this calculator fills in the rest: the missing sides and angles, the area, the perimeter, and whether it is equilateral, isosceles or scalene — and right, acute or obtuse.
Angles are shown in degrees. For three sides, the triangle must satisfy the triangle inequality (each side shorter than the sum of the other two) — otherwise no such triangle exists. Results are exact geometry, not estimates; rounding is to two decimals.
How it works
Three sides (SSS): area comes from Heron's formula — with semiperimeter s = (a + b + c) ÷ 2, area = √(s(s−a)(s−b)(s−c)). Each angle comes from the law of cosines, e.g. angle A = arccos((b² + c² − a²) ÷ 2bc). Base & height: area = ½ × base × height. Two sides & included angle (SAS): the third side comes from the law of cosines, c = √(a² + b² − 2ab·cos C), and area = ½ab·sin C. Worked example: sides 3, 4, 5: s = 6, so area = √(6 × 3 × 2 × 1) = 6, perimeter 12, and the angles are 36.87°, 53.13° and 90° — the classic right triangle, since 3² + 4² = 5².
Frequently asked questions
How do I know if three sides make a real triangle?
They must pass the triangle inequality: each side shorter than the sum of the other two. Sides 2, 3 and 9 fail (2 + 3 is not greater than 9), so no such triangle exists — the calculator says so rather than inventing angles.
Why is the 3-4-5 triangle special?
Because 3² + 4² = 9 + 16 = 25 = 5², it satisfies Pythagoras exactly, so the angle opposite the 5 side is exactly 90°. Builders have used a knotted 3-4-5 rope to lay out square corners for thousands of years.
What is the difference between isosceles and equilateral?
Equilateral means all three sides equal (and all angles 60°). Isosceles means exactly two sides equal — the angles opposite those sides are equal too. Scalene means no sides equal.
Can base and height alone give me the angles?
No. Many triangles share the same base and height but have different side lengths and angles — picture the apex sliding along a line parallel to the base. You need a side or an angle as well to pin the triangle down.
What does the included angle mean in the SAS mode?
It is the angle between the two sides you entered — the angle at the corner where they meet. If instead you know an angle that is not between your two sides, use the three-sides mode once you work out the third side, or the area formula ½ab·sin C still applies to any pair of sides and their included angle.