Compute the area of any triangle using A = ½ab·sin C (two sides + included angle), Heron's formula (three sides), or base × height. Also displays circumradius, inradius, altitudes, and triangle classification with a labeled diagram.
Inputs
Formula: A = ½ · a · b · sin C
Formula: s = (a+b+c)/2 | A = √(s(s−a)(s−b)(s−c))
Formula: A = ½ · b · h
Results
Deriving A = ½ab·sin C from Base × Height
h = b · sin C ⟹ A = ½ · base · h = ½ · a · (b·sin C)
In any triangle, drop a perpendicular from vertex B to side AC (the base a). The height of this perpendicular equals b · sin C, where C is the angle at vertex C between sides a and b.
Substituting into the familiar A = ½ · base · height gives A = ½ · a · b · sin C. This works for any included angle C — even obtuse angles — because sin(180° − C) = sin C.
Circumradius vs Inradius
R = abc / (4A) r = A / s
The circumradius R is the radius of the circle passing through all three vertices (circumscribed circle). By the Law of Sines: a/sin A = b/sin B = c/sin C = 2R.
The inradius r is the radius of the largest circle that fits inside the triangle (inscribed circle), tangent to all three sides. It equals the area divided by the semi-perimeter: r = A/s where s = (a+b+c)/2.
For a right triangle with legs p, q and hypotenuse c: R = c/2 and r = (p + q − c)/2.
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