Geometry
Enter three side lengths to find area, angles, perimeter, and triangle type.
Area
6.00 sq units
Scalene, right
A valid triangle requires: each side greater than zero, and the sum of any two sides greater than the third.
A triangle is a three-sided polygon. When you know the lengths of all three sides (a, b, c), you can compute its area, the three interior angles, and its type (equilateral, isosceles, or scalene) and shape (right, acute, or obtuse). This calculator does all that automatically.
Enter the three side lengths a, b, and c. The calculator first checks the triangle inequality—the sum of any two sides must be greater than the third, otherwise no triangle exists. If valid, it computes the area using Heron's formula, then finds each angle using the law of cosines, and finally classifies the triangle by type and shape.
Area = √[s(s−a)(s−b)(s−c)], where s = (a+b+c)/2. Angles: cos(A) = (b²+c²−a²)/(2bc), then A = arccos(result) in degrees.Heron's formula computes area from just the three sides without needing height. The law of cosines relates the sides to each angle, letting us find all three angles. Sides are labeled a, b, c and their opposite angles are A, B, C respectively.
| Input | Result | Notes |
|---|---|---|
| a=3, b=4, c=5 | 6 | A 3-4-5 right triangle: area is exactly 6 sq units (sides form 90°). |
| a=5, b=5, c=6 | 12 | An isosceles triangle with two equal sides of 5: area is about 12 sq units. |
| a=7, b=7, c=7 | 21.22 | An equilateral triangle with all sides 7: area is about 21.22 sq units. |
The triangle inequality states that the sum of any two sides must be greater than the third side. If this fails for any pair, the three lengths cannot form a triangle. For example, sides 1, 2, and 10 cannot form a triangle because 1 + 2 = 3, which is not greater than 10.
Heron's formula computes a triangle's area from only the three side lengths, without needing the height. It uses a semi-perimeter s = (a+b+c)/2, then area = √[s(s−a)(s−b)(s−c)]. It's elegant and avoids the need to draw altitudes.
The law of cosines relates a triangle's sides to its angles: c² = a² + b² − 2ab·cos(C). Rearranged to cos(C) = (a² + b² − c²)/(2ab), it lets you find any angle given the three sides. It's the generalization of the Pythagorean theorem.
Equilateral: all three sides equal (all angles 60°). Isosceles: exactly two sides equal. Scalene: all three sides different. The calculator tells you which type your triangle is.
Right: one angle is exactly 90°. Acute: all three angles are less than 90°. Obtuse: one angle is greater than 90°. The sum of all three angles is always 180°.
Yes. The calculator accepts any positive decimal numbers. For example, sides of 2.5, 3.7, and 4.2 are perfectly valid as long as they satisfy the triangle inequality.