AI Math Assistant
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About This Calculator
Triangles are the simplest polygon and the structural basis for all more complex shapes, making their properties foundational to geometry and engineering. This calculator determines missing sides, angles, area, and perimeter using the law of sines, law of cosines, and Heron's formula based on whatever measurements you provide. Structural engineers rely on triangular analysis because triangles are inherently rigid and cannot deform without changing side lengths.
Quick Tips
- 1 All interior angles of any triangle always sum to exactly 180 degrees.
- 2 Heron's formula finds area using only the three side lengths — no height needed.
- 3 The longest side must be shorter than the sum of the other two sides.
Example Calculation
A triangular sail with sides of 5m, 7m, and 9m.
Area: 17.41 m2 (Heron's formula) | Perimeter: 21m | Type: scalene acute
Heron's Formula for Triangle Area
Heron's formula calculates area from three sides without needing the height. First find the semi-perimeter s = (a+b+c)/2, then Area = √(s(s-a)(s-b)(s-c)). For example, a 3-4-5 triangle: s = 6, Area = √(6×3×2×1) = √36 = 6 square units.
Triangle Types by Sides and Angles
By sides: equilateral (all equal), isosceles (two equal), scalene (all different). By angles: acute (all angles < 90°), right (one angle = 90°), obtuse (one angle > 90°). You can determine the type from the sides using the Pythagorean relationship: if a² + b² = c², it is a right triangle.
Finding Angles from Sides
Using the law of cosines: cos(A) = (b² + c² - a²) / (2bc). This gives you angle A opposite to side a. Repeat for each angle. The angles must sum to 180°. This calculator computes all three angles automatically from the three sides you enter.
Triangle Inequality Theorem
The triangle inequality states that the sum of any two sides must be greater than the third side: a + b > c, a + c > b, b + c > a. If any of these conditions fails, the three lengths cannot form a valid triangle. The calculator checks this automatically.
Frequently Asked Questions
The sum of any two sides must be greater than the third side. For example, sides 2, 3, and 8 cannot form a triangle because 2 + 3 = 5, which is less than 8.
Compare the square of the longest side (c²) to the sum of squares of the other two (a² + b²). If equal: right triangle. If a² + b² > c²: acute. If a² + b² < c²: obtuse.
Yes, Heron's formula works for any valid triangle — right, acute, or obtuse. It only requires knowing all three side lengths, making it extremely versatile.
Both give the same result. The ½bh formula is simpler but requires knowing the height. Heron's formula is more general because it only needs the three sides. They are mathematically equivalent.