How this works
Two of the most common triangle questions handled in one tool. The "Area from sides" mode takes any three side lengths and uses Heron's formula to compute the area — no need to know the height or any angles. It also classifies the triangle (equilateral, isosceles, scalene, or right) and validates that the three sides actually form a valid triangle. The "Right-triangle" mode takes the two legs of a right triangle and returns the hypotenuse via the Pythagorean theorem, plus the area and perimeter for free.
The formula
a, b, c are the side lengths. s is the semi-perimeter (half the perimeter). For Heron's formula to return a real number, the three sides must satisfy the triangle inequality: each side must be less than the sum of the other two. The right-triangle mode treats a and b as the two legs and returns the hypotenuse — the longest side, opposite the right angle.
Example calculation
- A 3-4-5 right triangle. In Area mode, enter 3, 4, 5.
- Semi-perimeter s = 12 / 2 = 6. Area = √(6·3·2·1) = √36 = 6. Perimeter = 12. Type = right.
- In Right-triangle mode with legs 3 and 4: hypotenuse = √(9 + 16) = √25 = 5. Same triangle, different starting info.
Frequently asked questions
Why does the calculator say 'Invalid — sides do not form a triangle'?
Because of the triangle inequality: the sum of any two sides must be strictly greater than the third side. Try 1, 2, 5 and the longest side (5) is bigger than the other two added together (1 + 2 = 3). No real triangle can have those edge lengths — if you tried to lay out three sticks of those lengths end-to-end, they'd never meet.
Can I use Heron's formula with very long, thin triangles?
Yes, but watch out for floating-point precision when one side is nearly equal to the sum of the other two. The expression s(s − a)(s − b)(s − c) becomes a tiny positive number that may underflow to zero, giving a misleading area of 0. For very thin triangles, the more numerically stable form is √((a + b + c)(−a + b + c)(a − b + c)(a + b − c)) / 4. Our calculator uses the standard Heron form, which is fine for most everyday cases.
How do I find a triangle's area if I only know two sides and an angle?
Use the SAS formula: area = ½ · a · b · sin(C), where C is the angle between sides a and b. We don't expose this in the calculator yet (would need an angle-units toggle for degrees vs radians), but if you have the third side via the Law of Cosines (c² = a² + b² − 2ab·cos C), you can plug a, b, c into the area mode here.
Why is a 3-4-5 triangle classified as right and not scalene?
It's technically both — three different sides (so scalene) and one 90° angle (so right). The calculator picks the more specific label, since "right triangle" is the more useful piece of information when it applies. A 5-5-7 isosceles-and-not-right triangle would be labelled isosceles for the same reason.