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Triangle Calculator

A general Triangle Calculator solves any triangle from its three side lengths (SSS) using Heron's formula and the Law of Cosines. It verifies the Tria

Mathematics
Formula & Methodology
Semiperimeter: s = (a + b + c) ÷ 2 Heron's Area: Area = √[ s(s - a)(s - b)(s - c) ] Law of Cosines: cos(C) = (a² + b² - c²) ÷ (2ab)

To solve a triangle given its three side lengths (a, b, c): 1. Validate the Triangle Inequality Theorem: verify that a + b > c, a + c > b, and b + c > a. 2. Compute the semiperimeter s = (a + b + c) / 2. 3. Apply Heron's formula to evaluate area without needing the perpendicular altitude. 4. Calculate interior angles using the Law of Cosines.

Area via Heron's Formula (Scalene, Right)

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Overview

A general Triangle Calculator solves any triangle from its three side lengths (SSS) using Heron's formula and the Law of Cosines. It verifies the Triangle Inequality theorem, calculates area and perimeter, classifies the triangle, and determines all three interior angles.

Whether verifying carpentry frameworks, checking civil land boundaries, or solving trigonometry coursework, this comprehensive solver provides instant mathematical validation.

Worked Examples

Example 1: Classic 3-4-5 Right Triangle

Sides: a = 3, b = 4, c = 5

  1. Semiperimeter: (3 + 4 + 5) ÷ 2 = 6.
  2. Heron's formula: √[6(6-3)(6-4)(6-5)] = √[6 × 3 × 2 × 1] = √36 = 6.
  3. Angles: 36.87°, 53.13°, 90.00° (Right Triangle).
Final Answer: Area = 6.00, Perimeter = 12.00

Example 2: Equilateral Triangle (Sides 6, 6, 6)

Sides: a = 6, b = 6, c = 6

  1. Semiperimeter: 18 ÷ 2 = 9.
  2. Heron's: √[9 × 3 × 3 × 3] = √243 ≈ 15.5885.
  3. Angles: Exactly 60.00° each.
Final Answer: Area = 15.59

Example 3: Scalene Triangle (7, 8, 9)

Sides: a = 7, b = 8, c = 9

  1. s = 24 ÷ 2 = 12.
  2. Area: √[12(5)(4)(3)] = √720 ≈ 26.8328.
  3. Perimeter: 24.
Final Answer: Area = 26.83, Perimeter = 24.00

How to Use This Calculator

  1. Enter the lengths of side a, side b, and side c.
  2. Click "Calculate" to solve the triangle.
  3. Read the area, perimeter, classification, and all three interior angles.
  4. Click "Reset" to solve another triangle.

Reference Table

Sides (a, b, c)PerimeterHeron's AreaClassification
3, 4, 5126.00Scalene, Right
5, 12, 133030.00Scalene, Right
6, 6, 61815.59Equilateral
5, 5, 81812.00Isosceles
7, 8, 92426.83Scalene

Frequently Asked Questions

What is the Triangle Inequality Theorem?
The theorem states that for any valid triangle, the sum of any two sides must always be strictly greater than the third side. If a + b ≤ c, the lines cannot physically meet to form a closed triangle.
What is Heron's formula?
Heron's formula calculates the area of any triangle given only its three side lengths, eliminating the need to calculate altitudes or angles first.
Do the three calculated angles always sum to 180°?
Yes. In flat Euclidean planar geometry, the interior angles of any triangle always add up to exactly 180 degrees.
How do you tell if a triangle is a right triangle from its sides?
Check if the Pythagorean theorem holds: if a² + b² = c² (where c is the longest side), the triangle is a right triangle.
Can side lengths be negative or zero?
No. Side lengths must be positive real numbers greater than zero.

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Reviewed & Authored by: Dr. Elizabeth Vance, PhD Applied Mathematics · Last Updated: 2026-10-06
Computed via Heron's algebraic area formula and spherical trigonometry Law of Cosines.