Triangle Calculator
Unlike calculators built specifically for right triangles, this one solves any triangle shape from just its three side lengths — no right angle required. It finds the area, all three angles, the perimeter, and classifies the triangle’s type, all using the same three inputs.
How the Triangle Calculator Works
Enter the three side lengths (a, b, and c), and the calculator first verifies they can actually form a valid triangle, then computes area using Heron’s formula, all three interior angles using the law of cosines, the perimeter, and additional properties including the inradius and circumradius — the radii of the circles inscribed within and circumscribed around the triangle.
The Triangle Inequality: Why Not Every Three Lengths Form a Triangle
Before attempting any calculation, this calculator checks a fundamental geometric requirement: any two sides of a triangle must add up to more than the third side, or the three lengths simply cannot form a closed triangle at all — imagine trying to connect three rigid sticks where two of them combined are shorter than the third; they’d never meet to close the shape. If you enter three lengths that violate this rule, the calculator returns an error explaining the issue directly, rather than attempting a calculation that has no valid geometric answer.
Heron’s Formula: Finding Area From Sides Alone
Heron’s formula calculates a triangle’s area using only its three side lengths, with no need to know any angle or a separate height measurement — a genuinely elegant result, since area is usually thought of as requiring a base and a perpendicular height. The formula works by first finding the semi-perimeter (half the total perimeter), then combining it with each side length in a specific way: Area = √(s(s−a)(s−b)(s−c)), where s is the semi-perimeter. This calculator computes that formula directly, making it possible to find area from side lengths alone without needing to separately determine a height or angle first.
The Law of Cosines: Finding Angles From Sides Alone
Just as Heron’s formula finds area from sides alone, the law of cosines finds all three angles using only the three side lengths — a generalization of the Pythagorean theorem that works for any triangle, not just right triangles specifically. The law of cosines includes an extra term accounting for the angle between two sides, and when that angle happens to be exactly 90°, the formula’s extra term becomes zero, causing it to collapse back into the standard Pythagorean theorem exactly. This is precisely why the Pythagorean theorem only works for right triangles: it’s the law of cosines with that additional angle-dependent term removed, valid only in the specific case where that term equals zero.
Classifying Triangle Type: Right, Acute, or Obtuse
Once all three angles are known, this calculator classifies the triangle based on its largest angle: exactly 90° makes it a right triangle, greater than 90° makes it obtuse, and all angles under 90° makes it acute. This classification falls directly out of the same law of cosines calculation used to find the angles themselves — no separate check is needed, since the largest angle’s value alone determines which category the triangle falls into.
Inradius and Circumradius: Two Circles Every Triangle Has
Every triangle has exactly one circle that fits perfectly inside it, touching all three sides (radius = inradius), and exactly one circle that passes through all three of its corners (radius = circumradius) — both derived directly from the triangle’s area, perimeter, and side lengths. These values connect triangles and circles in a way that isn’t obvious at first glance, and this calculator computes both automatically alongside the other triangle properties. If you want to explore the resulting circle itself in more depth — converting between its radius, diameter, circumference, and area — the Circle Calculator handles that separate calculation once you have a radius value from this triangle calculator.
Right Triangles: A Simpler Special Case
If you already know your triangle has a right angle, the calculation simplifies considerably, since the Pythagorean theorem alone becomes sufficient without needing the fuller law of cosines this general calculator applies. The Right Triangle Calculator and Pythagorean Theorem Calculator handle that simpler, more direct case when a right angle is already guaranteed.
Flat Area Calculations for Other Shapes
If your shape isn’t a triangle at all, or you’re comparing a triangle’s area against other flat shapes like rectangles or circles, the Area Calculator covers several common flat shapes together, including a triangle option using the more basic base-and-height formula, distinct from this calculator’s side-lengths-only Heron’s formula approach.
Straight-Line Distance Between Coordinate Points
If you’re working with a triangle defined by coordinate points rather than known side lengths directly, you’d first need to calculate each side’s length from its coordinate pairs before using this calculator. The Distance Calculator handles that coordinate-to-length conversion directly, which then feeds into this triangle calculator once all three side lengths are known.
Frequently Asked Questions
Can any three lengths form a triangle?
No — the triangle inequality requires that any two sides added together must exceed the third side, or the lengths cannot form a closed triangle at all. This calculator checks that requirement automatically and returns an error if it isn’t satisfied.
How do you find a triangle’s area if you only know its three sides?
Use Heron’s formula: find the semi-perimeter (half the total perimeter), then calculate Area = √(s(s−a)(s−b)(s−c)), where s is the semi-perimeter and a, b, c are the three side lengths. This gives the area with no need for a separate height measurement.
How can you find a triangle’s angles knowing only its sides?
Use the law of cosines, which relates all three sides to any one angle. Solving that equation for each angle in turn gives all three interior angles using only the side lengths, without needing any angle to be known in advance.
What is the difference between the law of cosines and the Pythagorean theorem?
The Pythagorean theorem is a special case of the law of cosines that only applies when one angle is exactly 90°. The law of cosines works for any triangle by including an additional term for the angle between two sides, which becomes zero specifically in the right-triangle case.
What are inradius and circumradius?
Inradius is the radius of the largest circle that fits inside a triangle, touching all three sides. Circumradius is the radius of the circle that passes through all three of the triangle’s corners. Both are calculated directly from the triangle’s area, perimeter, and side lengths.