Triangle types and the triangle inequality
A triangle is classified by its side lengths and by its largest angle; the table below summarizes both systems.
| Classification basis | Type | Description |
|---|---|---|
| By sides | Equilateral | All three sides equal; all angles 60° |
| By sides | Isosceles | Exactly two sides equal |
| By sides | Scalene | All three sides different lengths |
| By angle | Acute | All three angles less than 90° |
| By angle | Right | One angle exactly 90° |
| By angle | Obtuse | One angle greater than 90° |
- The triangle inequality theorem requires that the sum of any two sides exceed the third side (a + b > c, a + c > b, b + c > a); three lengths that fail this cannot form a triangle, and the calculator returns no result in that case.
- In "two sides + included angle" mode, the entered angle C must lie strictly between 0° and 180°, since 0° or 180° would collapse the triangle into a straight line.
- The three interior angles of any triangle always sum to exactly 180°, which the calculator uses as a check when deriving the third angle.
What does this triangle calculator solve?
Any triangle is fully determined by three independent measurements, most commonly three sides (SSS), two sides and the angle between them (SAS), or — for a right triangle — its two perpendicular legs. From any of these, every other side, angle, the area and the perimeter can be derived using standard trigonometric identities.
This calculator supports all three input modes. In "three sides" mode it validates the triangle inequality (each side must be shorter than the sum of the other two) before solving, since three lengths that fail this test cannot form a triangle.
How to use this triangle calculator
- Choose the known-values mode: three sides, two sides plus an included angle, or right-triangle legs.
- For "three sides", enter side a, side b and side c.
- For "two sides + included angle", enter side a, side b and the angle C between them (0°–180°, exclusive).
- For "right triangle", enter the two legs as side a and side b; the hypotenuse is derived automatically.
- Read the area, perimeter, the missing side (if applicable) and all three interior angles, which update instantly.
The formulas behind triangle solving
Heron's formula computes a triangle's area directly from its three side lengths, without needing to know any angle. The law of cosines both derives a missing side (SAS mode) and recovers all three angles from three known side lengths.
Common mistakes
- Entering three side lengths that violate the triangle inequality (for example 2, 3 and 10), which cannot form a valid triangle.
- In SAS mode, entering the angle that is not actually between the two given sides — the law of cosines requires the included angle.
- Confusing a right triangle's legs with its hypotenuse when using "right triangle" mode — sides a and b must be the two perpendicular legs, not the hypotenuse.
- Mixing degrees and radians when reading the angle results, which are always returned in degrees.
Questions fréquentes
How do you find the area of a triangle with three sides?
Use Heron's formula: A = √(s(s−a)(s−b)(s−c)), where s is the semi-perimeter (a+b+c)/2. For a 13-14-15 triangle, s = 21, and A = √(21×8×7×6) = √7056 = 84 square units exactly.
What are the angles of a 13-14-15 triangle?
For sides a = 13, b = 14, c = 15, the law of cosines gives angle A ≈ 53.13°, angle B ≈ 59.49° and angle C ≈ 67.38°, which sum to 180° (allowing for rounding).
How do you solve a triangle with two sides and an included angle?
Use the law of cosines to find the third side: c² = a² + b² − 2ab·cos C. Then find the remaining two angles using the law of cosines rearranged for angles, or the law of sines.
What is the triangle inequality?
The triangle inequality states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. If this fails for any pair, the three lengths cannot form a triangle.
How do you find the hypotenuse of a right triangle?
The hypotenuse equals the square root of the sum of the squares of the two legs: c = √(a² + b²). A right triangle with legs 3 and 4 has a hypotenuse of √(9+16) = √25 = 5.
Can a triangle have two right angles?
No. Since the three interior angles of a planar triangle always sum to exactly 180°, having two 90° angles would leave 0° for the third angle, which is not a valid triangle.
Références
- Weisstein, Eric W. "Heron's Formula" and "Law of Cosines." MathWorld — A Wolfram Web Resource.
- Standard trigonometry textbook conventions (e.g. Larson, Trigonometry, Cengage Learning).