Heron's Formula Triangle Calculator

Area
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Perimeter
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Semiperimeter (s)
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Interactive Triangle Visualization

About Heron's Formula

Heron's formula (also known as Hero's formula) is a mathematical equation discovered by Hero of Alexandria around 60 AD. It provides an elegant way to calculate the area of a triangle when you know the lengths of all three sides, without needing to know any angles or heights.

The Formula

Area = √[s(s-a)(s-b)(s-c)]

where s is the semiperimeter: s = (a + b + c) / 2

Why is Heron's Formula Useful?

  • No angles needed: Calculate area directly from side lengths without trigonometry
  • Works for all triangles: Applies to acute, obtuse, right, scalene, isosceles, and equilateral triangles
  • Surveying & navigation: Essential for land measurement when only distances are known
  • Computer graphics: Used in 3D rendering and mesh calculations
  • Engineering: Structural analysis and design applications

Step-by-Step Example

Problem: Find the area of a triangle with sides a = 5, b = 6, c = 7

Step 1: Calculate the semiperimeter

s = (a + b + c) / 2 = (5 + 6 + 7) / 2 = 18 / 2 = 9

Step 2: Apply Heron's formula

Area = √[s(s-a)(s-b)(s-c)]
Area = √[9(9-5)(9-6)(9-7)]
Area = √[9 × 4 × 3 × 2]
Area = √[216]
Area = 14.696 square units

Special Cases

Right Triangle (3-4-5)

s = (3+4+5)/2 = 6

Area = √[6×3×2×1] = √36 = 6

Verification: (1/2)×base×height = (1/2)×3×4 = 6 ✓

Equilateral Triangle (side = 6)

s = (6+6+6)/2 = 9

Area = √[9×3×3×3] = √243 = 15.588

Formula: (√3/4)×a² = 15.588 ✓

Isosceles Triangle (5-5-6)

s = (5+5+6)/2 = 8

Area = √[8×3×3×2] = √144 = 12

Perfect square result!

Important Notes

Triangle Inequality Theorem: For three sides to form a valid triangle, the sum of any two sides must be greater than the third side. Otherwise, the sides cannot connect to form a closed shape.

• a + b > c
• a + c > b
• b + c > a

Example of invalid triangle: Sides 2, 3, 10 cannot form a triangle because 2 + 3 = 5, which is not greater than 10.

Historical Context

Hero of Alexandria (c. 10–70 AD) was a Greek mathematician and engineer who made significant contributions to geometry, mechanics, and pneumatics. While the formula bears his name, some historians believe it may have been known earlier. Hero's work "Metrica" documented this formula along with methods for calculating areas and volumes of various geometric shapes.

Related Calculations

Once you have the triangle's area from Heron's formula, you can calculate many other properties:

  • Heights (altitudes): h = 2×Area/base
  • Inradius: r = Area/s (radius of inscribed circle)
  • Circumradius: R = (abc)/(4×Area) (radius of circumscribed circle)
  • Angles: Use Law of Cosines: cos(A) = (b²+c²-a²)/(2bc)
  • Medians: ma = √(2b²+2c²-a²)/2

Try it yourself: Use the calculator above to experiment with different triangle side lengths. Try the preset buttons for common triangles, or enter your own values to see all the calculated properties including area, angles, heights, and classification!

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