Triangle Area Calculator

Pick by what you actually measured. If you have one side (the base) and the perpendicular distance from that side to the opposite vertex (the height), use A = (1/2) base x height. This is the simplest and the fastest, but the height has to be perpendicular to the base, not the length of a tilted side. If you have two sides and the angle between them (an SAS measurement), use A = (1/2) a b sin(C). And if you have all three sides but no angles or heights, use Heron's formula A = sqrt(s(s-a)(s-b)(s-c)) with s = (a+b+c)/2. All three give the same area for the same triangle, so pick the one that needs the fewest extra steps from your inputs.

It is named after Hero of Alexandria, a Greek mathematician and engineer who described the result in his work Metrica around 60 AD. Hero's proof used geometry, not algebra, and worked with line segments instead of variables. There is some evidence Archimedes knew the formula nearly three centuries earlier (a related form shows up in Arabic translations of his lost work), but it is Hero's exposition that survived in continuous use. The formula's value is that it solves the area of a triangle from side lengths alone, no angles and no altitudes needed, which is rare among geometric formulas and made it indispensable for land surveying in the ancient world.

For any three positive lengths to form a triangle, each side must be strictly shorter than the sum of the other two: a + b > c, b + c > a, and a + c > b. If any one inequality is violated, the three lengths simply cannot meet at three vertices in a plane. For example, 1, 2, 5 fails because 1 + 2 = 3, which is less than 5: the two short sides cannot reach across the long one. The boundary case (a + b = c) is degenerate, the three points lie on a single line, so no proper triangle exists. The Heron mode of this calculator checks all three inequalities before computing area; if they fail, it returns isValidTriangle = false instead of a nonsense answer.

Side classification looks at how many sides are equal. Equilateral means all three sides are equal (and as a consequence all three angles are 60 degrees); isosceles means exactly two sides are equal; scalene means all three sides differ. Angle classification looks at the largest interior angle. Right means the largest angle is exactly 90 degrees; acute means the largest angle is under 90; obtuse means the largest is over 90 (and under 180). The two classifications are independent, so a triangle can be both isosceles and right (a 45-45-90), scalene and obtuse (a 3-4-6), equilateral and acute (60-60-60), or any other combination.

The formula A = (1/2) a b sin(C) measures the area of the parallelogram with sides a and b and angle C between them, then halves it. The C in that formula is specifically the angle where sides a and b meet. If you plug in a different angle (say the angle opposite side a), the trig identity falls apart because sin of an unrelated angle has no geometric tie to how a and b are arranged. The reason the included angle works is that the height of the triangle from side b is exactly a sin(C), and that height drops perpendicularly onto b only when C is the angle between them. If you have one side and an angle that is not included between two known sides, you cannot use SAS directly; you need a different setup (ASA, AAS, or solving the triangle first with the law of sines).