Three sides, the whole triangle.
Enter the three side lengths and see the area by Heron's formula, the perimeter, all three angles, the heights, and what type of triangle you have, drawn to scale as you type.
The three sides
Side a is opposite angle A, side b opposite B, side c opposite C. Any two sides must add up to more than the third.
What this triangle tells you
Triangle scorecard
Angles and heights
Each angle sits opposite its matching side. The height (altitude) to a side is 2 x Area divided by that side.
| Vertex | Angle | Opposite side | Side length | Height to that side |
|---|
Triangle geometry, explained
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.
Common questions
How do I know if three sides can form a triangle?
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.
What makes a triangle right, acute, or obtuse?
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.
Can Heron's formula work for all triangles?
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.
How is the area formula ½ × base × height related to Heron's formula?
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.
Results are computed from the three side lengths you enter. Values are rounded for display, so recomputed figures may differ in the last decimal.