Triangle Calculator

How to Use This Calculator

Enter all three side lengths (a, b, c). The calculator uses Heron's formula for area, the Law of Cosines for angles, and classifies the triangle type.

Formula

Example

Frequently Asked Questions

• How do I calculate triangle area from three sides? ▼
  Use Heron's formula: first compute the semi-perimeter s = (a + b + c) / 2, then Area = √(s(s−a)(s−b)(s−c)). For example, for a triangle with sides 3, 4, 5: s = 6, Area = √(6×3×2×1) = √36 = 6 square units. This formula works for any triangle when all three sides are known, without needing to know the height. It is equivalent to the more familiar formula Area = ½ × base × height, but Heron's does not require you to calculate the height separately.
• How do I find the angles of a triangle from its sides? ▼
  Use the Law of Cosines: cos A = (b² + c² − a²) / (2bc), where A is the angle opposite side a. Then A = arccos((b² + c² − a²) / (2bc)). Once you have two angles, the third is 180° − A − B (since all angles in a triangle sum to 180°). Example: sides 5, 7, 8. For angle A opposite side 5: cos A = (49 + 64 − 25) / (2×7×8) = 88/112 ≈ 0.786, so A ≈ 38.2°. Repeat for the other angles.
• What is the triangle inequality? ▼
  The triangle inequality states that the sum of any two sides of a triangle must be strictly greater than the third side. In other words, for sides a, b, c: a + b > c, a + c > b, and b + c > a must all hold. If any condition fails, the three lengths cannot form a closed triangle. For example, sides 1, 2, 10 are invalid because 1 + 2 = 3 < 10. The calculator checks this automatically and reports "Invalid triangle" if the input violates the inequality.
• What types of triangles does this handle? ▼
  The calculator handles all triangle types: equilateral (all three sides equal, all angles 60°), isosceles (two sides equal, two base angles equal), scalene (all sides different, all angles different), right (one 90° angle, satisfies a² + b² = c²), acute (all angles less than 90°), and obtuse (one angle greater than 90°). The calculator identifies the type automatically based on the computed angles and side relationships.
• What is a right triangle? ▼
  A right triangle has exactly one 90-degree angle. The side opposite the right angle is called the hypotenuse (c), the longest side. The other two sides are the legs (a and b). Right triangles satisfy the Pythagorean theorem: a² + b² = c². Famous right triangles include the 3-4-5 triple (9+16=25) and the 5-12-13 triple. The two non-right angles in a right triangle are complementary (they sum to 90°). Right triangles are fundamental in trigonometry: sin(A) = a/c, cos(A) = b/c, tan(A) = a/b.

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