Calculate Triangle Area

Calculate triangle area from three sides using Heron's formula

Heron's Formula Calculator

Heron's Area Formula

Heron's formula calculates the area from three side lengths without angles or heights needed.

First side length
Second side length
Third side length
Calculation Results
Area A:
Height h:
Angle α:
Angle β:
Angle γ:
Incircle Radius:
Incircle Area:

Visualization

Triangle with incircle using Heron

The diagram shows a triangle with all three sides a, b, c and the incircle.
Heron's formula calculates the area from side lengths only.

Note Triangle Inequality

Each side must be smaller than the sum of the other two: a < b + c, b < a + c, c < a + b

What is Heron's Formula?

Heron's formula is one of the most elegant methods in geometry:

  • Only side lengths: No angles or heights required
  • Universal: Works for all triangle types
  • Complete: Determines all other triangle properties
  • Historical: Named after Hero of Alexandria (1st century AD)
  • Applications: Surveying, CAD, material calculation
  • Foundation: For further calculations like incircle and angles

The Triangle's Incircle

The incircle is the largest circle that fits completely inside the triangle:

Incircle Properties
  • Touches all three sides of the triangle
  • Center lies at intersection of angle bisectors
  • Radius r = Area / Semi-perimeter
  • Optimal use of triangle's interior space
Calculation via Heron
  • Area A via Heron's formula
  • Semi-perimeter s = (a + b + c) / 2
  • Incircle radius r = A / s
  • Incircle area = π × r²

Understanding Calculation Steps

The Heron calculation follows systematic steps:

1. Validation

Check triangle inequality for all three sides

2. Semi-perimeter

s = (a + b + c) / 2 as foundation for all further calculations

3. Heron's Formula

A = √[s(s-a)(s-b)(s-c)] for area calculation

4. Further Values

Angles via law of cosines, heights and incircle properties

Heron's Formula and Further Calculations

The Classic Heron's Formula
\[A = \sqrt{s \cdot (s-a) \cdot (s-b) \cdot (s-c)}\]
\[s = \frac{a + b + c}{2}\] (Semi-perimeter)
Alternative Heron's Formulas
\[A = \frac{1}{4}\sqrt{(a+b+c)(-a+b+c)(a-b+c)(a+b-c)}\]
\[A = \frac{1}{4}\sqrt{4a^2b^2-(a^2+b^2-c^2)^2}\]
Incircle Calculations
\[r = \frac{A}{s} = \frac{\sqrt{s(s-a)(s-b)(s-c)}}{s}\]
\[A_{Incircle} = \pi r^2\]
Angle α
\[\alpha = \arccos\left(\frac{b^2+c^2-a^2}{2bc}\right)\]
Angle β
\[\beta = \arccos\left(\frac{a^2+c^2-b^2}{2ac}\right)\]
Angle γ
\[\gamma = \arccos\left(\frac{a^2+b^2-c^2}{2ab}\right)\]
Calculate Height
\[h_a = \frac{2A}{a} = \frac{2\sqrt{s(s-a)(s-b)(s-c)}}{a}\]

Height to side a

Circumcircle Radius
\[R = \frac{abc}{4A}\]

Radius of the circumcircle

Symbols and Notation
  • A: Area of the triangle
  • a, b, c: Side lengths of the triangle
  • s: Semi-perimeter (a+b+c)/2
  • α, β, γ: Interior angles of the triangle
  • r: Incircle radius
  • R: Circumcircle radius
  • h: Height of the triangle
  • √: Square root (characteristic of Heron)

Calculation Example

Given
a = 5 b = 4 c = 3
1. Semi-perimeter
\[s = \frac{5 + 4 + 3}{2} = 6\]

Semi-perimeter s = 6

2. Heron's Formula
\[A = \sqrt{6 \cdot 1 \cdot 2 \cdot 3}\] \[A = \sqrt{36} = 6\]

The area is 6 square units

3. Incircle
\[r = \frac{6}{6} = 1\] \[A_{Incircle} = \pi \cdot 1^2 = \pi\]

Incircle radius r = 1, area = π ≈ 3.14

4. Special Property

This is a right triangle!
3² + 4² = 9 + 16 = 25 = 5²
Check: A = ½ × 3 × 4 = 6 ✓

5. Angles
α = 90° (right angle)
β ≈ 53.13°
γ ≈ 36.87°

Heron's Formula in Mathematics and Applications

Heron's formula (also called Hero's area formula) is one of the most remarkable discoveries of ancient mathematics. It enables the calculation of triangle area exclusively from the three side lengths, without requiring angles, heights, or other geometric constructions.

Historical Background

The formula bears the name of the Alexandrian mathematician Hero (ca. 10-70 AD):

  • Hero of Alexandria: Greek mathematician and engineer of antiquity
  • Documentation: First systematic presentation in "Metrica" (ca. 60 AD)
  • Earlier origins: Possibly already known to Archimedes (287-212 BC)
  • Chinese mathematics: Similar formulas in "Liu Hui" (3rd century AD)
  • Modern significance: Fundamental building block of analytical geometry

Mathematical Structure and Elegance

Symmetric Form

The formula A = √[s(s-a)(s-b)(s-c)] shows perfect symmetry with respect to all three side lengths. Each side is treated equally.

Square Root Character

The characteristic root makes the formula unique among area formulas and gives it its special mathematical elegance.

Semi-perimeter as Key

The semi-perimeter s = (a+b+c)/2 is the central parameter that encodes all geometric properties of the triangle.

Product Form

The product s(s-a)(s-b)(s-c) contains all information about the shape and size of the triangle in compact form.

Geometric Interpretation

The terms (s-a), (s-b), (s-c) have an intuitive geometric meaning:

  • s-a: "Excess" of semi-perimeter over side a
  • s-b: "Excess" of semi-perimeter over side b
  • s-c: "Excess" of semi-perimeter over side c
  • Product: Encodes the "balance" of side ratios

Practical Applications

Heron's formula finds wide application in various fields:

  • Land surveying: Area calculation of irregular properties
  • Construction planning: Material calculation for triangular constructions
  • CAD software: Automatic area calculation in technical drawings
  • Geodesy: Triangulation and terrain surveying
  • Computer graphics: Polygon area calculation and mesh analysis
  • Physics: Cross-section calculations and materials science

Related Calculations

Heron's formula is the starting point for many other triangle calculations:

Incircle Properties
  • Incircle radius: r = A/s
  • Incircle area: π × r²
  • Tangent points on sides
  • Optimal circle inscription
Other Properties
  • Heights: h = 2A/side
  • Angles via law of cosines
  • Circumcircle radius: R = abc/(4A)
  • Centroid and other centers

Alternative Formulations

Heron's formula can be presented in various mathematically equivalent forms:

  • Classic form: A = √[s(s-a)(s-b)(s-c)]
  • Extended form: A = ¼√[(a+b+c)(-a+b+c)(a-b+c)(a+b-c)]
  • Determinant form: A = ¼√[4a²b² - (a²+b²-c²)²]
  • Brahmagupta generalization: For quadrilaterals (for cyclic quadrilaterals)

Numerical Aspects

In practical application, various numerical considerations are important:

  • Triangle inequality: Mandatory check before calculation
  • Numerical stability: Problems with very "thin" triangles
  • Rounding errors: Amplification with unfavorable side ratios
  • Alternative algorithms: Kahan formula for better numerical properties

Generalizations and Extensions

Higher Dimensions

Generalizations for tetrahedra (Cayley-Menger determinant) and higher-dimensional simplices in analytical geometry.

Spherical Geometry

Adaptations for triangles on sphere surfaces with correspondingly modified trigonometric relationships.

Hyperbolic Geometry

Variants for non-Euclidean geometries with adapted distance concepts and angle definitions.

Computer Algebra

Symbolic calculations and exact arithmetic for theoretical investigations and proofs.

Modern Significance

In today's mathematics and technology, Heron's formula remains highly relevant:

  • Algorithmic geometry: Building block for computational geometry
  • Finite elements: Area calculation in numerical simulations
  • Computer vision: Object recognition and 3D reconstruction
  • Robotics: Path planning and collision detection
  • Game development: Physics engines and collision detection
Summary

Heron's formula is a masterpiece of ancient mathematics that has lost none of its elegance and practical significance to this day. It connects mathematical beauty with practical applicability and forms a bridge between classical geometry and modern computer-aided calculations. Its simplicity in application combined with mathematical depth makes it an indispensable tool for mathematicians, engineers, and scientists of all disciplines.

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Bisector of a triangle  •  Equilateral triangle  •  Right triangles  •  Right triangle, given 1 side and 1 angle  •  Isosceles right triangles  •  Isosceles triangles  •  Triangle area, given 2 sides and 1 anglee  •  Triangle area, given 1 side and 2 angles  •  Triangle, Incircle, given 3 sides  •  Area of a triangle given base and height  •  Triangle vertices, 3 x/y points  •