Enter the three side lengths of a triangle to calculate all angles, area, perimeter, and classify the triangle type.
This tool calculates missing sides, angles, area, and perimeter of a triangle based on the values you already know, using trigonometric rules and geometric formulas. It's useful for geometry homework, construction layout, and any calculation involving triangular shapes.
Depending on the known values, the calculator applies the Law of Sines (a/sin A = b/sin B = c/sin C), the Law of Cosines (c² = a² + b² − 2ab·cos C), or basic angle sum rules (angles in a triangle always total 180°). Area is commonly calculated as ½ × base × height, or using Heron's formula when all three sides are known.
Enter the known sides and/or angles of your triangle (at least three values, including at least one side), and the calculator solves for the remaining sides, angles, area, and perimeter.
Example: A triangle with sides 5, 6, and 7 (all sides known) has an area of approximately 14.7 square units, calculated using Heron's formula.
This is a fundamental property of Euclidean geometry — for any triangle on a flat plane, the sum of its three interior angles is always exactly 180 degrees.
It relates the sides and angles of any triangle (not just right triangles), useful when you know two angles and one side, or two sides and a non-included angle.
It's used to find a missing side or angle when you know either all three sides, or two sides and the included angle between them — it generalizes the Pythagorean theorem to non-right triangles.
It calculates a triangle's area directly from its three side lengths, without needing to know any angle, using the formula Area = √(s(s-a)(s-b)(s-c)), where s is the semi-perimeter.
Not always — knowing three angles only determines the triangle's shape, not its size, since infinitely many similar triangles share the same angles; at least one side length is needed to fully define a specific triangle.
A triangle calculator typically works out missing sides, angles, area, and perimeter of a triangle based on whatever information you already know — for example, given two sides and the angle between them, or all three side lengths, or two angles and one side. Behind the scenes it applies foundational geometric principles like the Pythagorean theorem for right triangles, the Law of Sines, and the Law of Cosines for more general triangles, saving you from manually working through trigonometric formulas and reducing the chance of an arithmetic slip.
Triangles are classified by their side lengths as equilateral, isosceles, or scalene, and by their angles as acute, right, or obtuse, and knowing which type you're working with often determines which formula applies most directly. A right triangle, for instance, allows the straightforward use of the Pythagorean theorem, while a triangle with no right angle requires the more general Law of Sines or Law of Cosines, which is exactly the kind of calculation this tool automates.
Triangle geometry isn't just a classroom exercise — it's the foundation of fields like architecture, engineering, and navigation, where calculating precise angles and distances is essential for everything from designing a stable roof truss to determining the height of a tall structure using angle measurements taken from the ground. Surveyors use triangulation to measure land and distances that can't be directly measured, and even everyday tasks like figuring out how much material you need for a triangular piece of a project benefit from quick, accurate calculations.
Triangle calculations can be done using either degrees or radians for angle measurements, and mixing the two up is a common source of error, since a calculation expecting radians will produce a wildly incorrect result if fed a degree value directly, or vice versa. This calculator handles the necessary unit consistency internally, so you can work in whichever angle unit you're most comfortable with without needing to manually convert first.
Not every combination of three side lengths actually forms a valid triangle — the triangle inequality theorem states that the sum of any two sides must be greater than the third side, and if this condition isn't met, no triangle can exist with those measurements. This calculator automatically checks for this before attempting to solve, helping catch input errors early rather than producing a nonsensical result from an impossible set of side lengths.
Trigonometric calculations often produce results with several decimal places, and how much you round depends on context — a construction project may need precision to a fraction of an inch, while a rough estimate for a school assignment can typically be rounded to one or two decimal places.
How is the area of any triangle calculated when it's not a right triangle? Heron's formula calculates area using all three side lengths without needing to know the height directly, providing a versatile method that works for any triangle shape, not just right triangles where the simpler base-times-height formula applies.
Why do the three angles of any triangle always sum to 180 degrees? This is a fundamental geometric property following from Euclidean geometry, holding true for every possible triangle regardless of its specific shape or size, making it a reliable tool for solving for an unknown angle when the other two are known.