# Triangle Inequality Calculator

Check if three sides form a valid triangle using the triangle inequality theorem. Classify by sides and angles. Calculate area and perimeter.

## What this calculates

Determine whether three side lengths can form a valid triangle, classify it by sides (equilateral, isosceles, scalene) and angles (acute, right, obtuse), and compute the area and perimeter.

## Inputs

- **Side a** — min 0 — Length of side a.
- **Side b** — min 0 — Length of side b.
- **Side c** — min 0 — Length of side c.

## Outputs

- **Valid Triangle?** — formatted as text — Whether the three sides satisfy the triangle inequality theorem.
- **Type by Sides** — formatted as text — Equilateral (all equal), isosceles (two equal), or scalene (all different).
- **Type by Angles** — formatted as text — Acute (all angles < 90°), right (one angle = 90°), or obtuse (one angle > 90°).
- **Area** — Area calculated using Heron's formula.
- **Perimeter** — Sum of all three sides.

## Details

The triangle inequality theorem states that the sum of any two sides of a triangle must be greater than the third side. If a + b > c, a + c > b, and b + c > a all hold, the three lengths form a valid triangle. If any condition fails, no triangle is possible.

Triangles are classified by sides: equilateral (all three sides equal), isosceles (exactly two sides equal), and scalene (all sides different). They are also classified by angles: acute (all angles less than 90°), right (one angle exactly 90°), and obtuse (one angle greater than 90°). The angle classification can be determined without computing angles by comparing c² (the longest side squared) with a² + b²: if equal, it is right; if less, acute; if greater, obtuse.

The area of a triangle when all three sides are known is calculated using Heron's formula: A = √(s(s-a)(s-b)(s-c)), where s = (a + b + c) / 2 is the semi-perimeter. This elegant formula avoids needing to know any angles.

## Frequently Asked Questions

**Q: What is the triangle inequality theorem?**

A: The triangle inequality theorem states that for any triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side. This must hold for all three combinations: a+b > c, a+c > b, and b+c > a.

**Q: How do I determine if a triangle is acute, right, or obtuse?**

A: Sort the sides so c is the longest. If a² + b² = c², it is a right triangle. If a² + b² > c², it is acute (all angles < 90°). If a² + b² < c², it is obtuse (the angle opposite c is > 90°).

**Q: What is Heron's formula?**

A: Heron's formula calculates the area of a triangle from its three sides: A = √(s(s-a)(s-b)(s-c)), where s = (a+b+c)/2 is the semi-perimeter. It works for any triangle without needing to know the angles or height.

**Q: Can a triangle have sides 1, 2, 3?**

A: No. The triangle inequality requires 1 + 2 > 3, but 1 + 2 = 3 (not strictly greater). These sides form a degenerate triangle (a straight line with zero area). Each pair of sides must sum to strictly more than the third.

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Source: https://vastcalc.com/calculators/math/triangle-inequality
Category: Math
Last updated: 2026-04-21