Triangle calculator (three sides)

Area = √(s(s − a)(s − b)(s − c)) with s = (a + b + c) ÷ 2 (Heron's formula). Angle A = arccos((b² + c² − a²) ÷ 2bc), from the law of cosines. Three sides fix a triangle completely, as long as each side is shorter than the other two added together. The area comes from Heron's formula, arranged so it stays accurate for long, thin triangles, and each angle comes from the law of cosines. The calculator also names the triangle by its sides (equilateral, isosceles or scalene) and by its largest angle (acute, right or obtuse).

With side a 7, side b 8, side c 9, the area is 26.83281573. Perimeter: 24; Angle A (opposite side a): 48.1896851 °; Angle B (opposite side b): 58.41186449 °; Angle C (opposite side c): 73.3984504 °; Triangle type: Scalene, acute; Inradius (inscribed circle): 2.236067977; Circumradius (circumscribed circle): 4.695742753.

This is an estimate based on the inputs and assumptions shown on this page, not professional advice. Check anything with financial, health or safety consequences independently.

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Each side must be shorter than the other two combined. With sides 2, 3 and 6, for example, 2 + 3 is less than 6, so the two short sides cannot meet. If the sum exactly equals the third side the triangle collapses into a straight line with zero area.

It gives the area of a triangle from its three sides alone. Halve the perimeter to get s, then the area is the square root of s(s − a)(s − b)(s − c). A 3-4-5 triangle has s = 6 and area √(6 × 3 × 2 × 1) = 6.

Square the two shorter sides and add them. If the total equals the square of the longest side, the triangle has a right angle; if it is smaller, the largest angle is obtuse; if it is larger, every angle is acute.