Triangle Calculator
Enter the three side lengths of a triangle to find its area and all three angles.
How this calculator works
The area is calculated using Heron's formula from the three side lengths, and each angle is found using the law of cosines.
If the three sides can't form a valid triangle, no result is shown.
The triangle inequality theorem
For three lengths to form a valid triangle, the sum of any two sides must be greater than the third side — this is called the triangle inequality theorem. If this condition isn't met, no triangle can be formed with those measurements, which is why this calculator flags invalid combinations.
This matters in real-world applications like construction and design, where checking the triangle inequality before finalizing measurements can catch errors before they become costly mistakes. For a deeper walkthrough of the formulas used here, see our triangle calculator guide.
A worked example
For a triangle with sides 5, 6, and 7: using Heron's formula, s equals 9, and area equals the square root of (9 times 4 times 3 times 2), or the square root of 216, approximately 14.7 square units.
The angles work out to approximately 44.4, 57.1, and 78.5 degrees (summing to 180 as expected) — calculated using the law of cosines from the three known side lengths.
Types of triangles by angle and side
Triangles can be classified by their angles (acute — all angles under 90°, right — one 90° angle, or obtuse — one angle over 90°) and by their sides (equilateral — all sides equal, isosceles — two sides equal, or scalene — all sides different). This calculator's angle outputs let you determine which category your triangle falls into.
The three angles of any triangle always sum to exactly 180 degrees, regardless of the triangle's size or shape — a fundamental geometric fact this calculator relies on when computing the third angle from the other two. If you need sine, cosine, or tangent of an angle directly, the Trigonometry Calculator handles that.
Frequently asked questions
What if my three sides can't form a triangle?
Any side must be shorter than the sum of the other two — if that's not true, no valid triangle exists and the calculator won't return a result.
What is Heron's formula used for?
It calculates a triangle's area directly from its three side lengths, without needing to know any angles first.
Why did my calculator say 'invalid triangle'?
This happens when one side is longer than or equal to the sum of the other two sides — check the triangle inequality theorem, or double-check your measurements for typos.
Can I calculate a triangle's area with just two sides and an angle?
Yes, using a different formula (½ × a × b × sin(C)) — this calculator specifically uses three known sides (Heron's formula) rather than the side-angle-side method.
Do the three angles always add up to 180 degrees?
Yes, for any triangle in standard Euclidean geometry — this is a fundamental property used to verify or calculate the third angle when two are known.
How do you find the angle of a triangle?
With all three sides known (SSS), use the Law of Cosines: cos(A) = (b² + c² - a²) / 2bc. With two sides and the included angle (SAS), use the Law of Cosines for the third side, then Law of Sines for the other angles. Remember: all three angles must sum to 180°.
How do you find the missing side of a triangle?
For right triangles: use the Pythagorean theorem (a² + b² = c²). For any triangle with a known side and two angles: use the Law of Sines: a/sin(A) = b/sin(B) = c/sin(C). For SAS triangles: use the Law of Cosines: c² = a² + b² - 2ab·cos(C).
What is the formula for the area of a triangle?
Area = ½ × base × height. If you know two sides and the included angle: Area = ½ × a × b × sin(C). If you know all three sides (Heron's formula): Area = √(s(s-a)(s-b)(s-c)) where s = (a+b+c)/2.