Last updated: July 31, 2026
Heron's Formula Calculator
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Formula
s = (a+b+c)/2, Area = √(s(s-a)(s-b)(s-c))
Where:
- a=Side a(units)
- b=Side b(units)
- c=Side c(units)
- s=Semiperimeter(units)
- A=Triangle Area(square units)
Worked Examples
Classic 13-14-15 triangle
A well-known triangle with integer area.
- 1s = (13+14+15)/2 = 21
- 2A = √(21×8×7×6) = √7056 = 84
Right triangle 3-4-5
Pythagorean triple — area matches ½×base×height.
- 1s = (3+4+5)/2 = 6
- 2A = √(6×3×2×1) = √36 = 6
Equilateral triangle side 10
All sides equal, area = 25√3.
- 1s = 15
- 2A = √(15×5×5×5) = √1875 = 25√3 ≈ 43.301270
Invalid triangle 1-2-10
Triangle inequality violation.
- 11+2=3 < 10 → triangle inequality fails → error
Introduction
Heron's formula finds the exact area of a triangle from its three side lengths alone. It is especially useful when you know all three sides but do not know a perpendicular height. This calculator also returns semiperimeter, perimeter, and the altitude to each side for deeper triangle analysis.
Heron's Formula Step by Step
The method works in two stages: first compute semiperimeter, then use the square-root expression for area.
Semiperimeter: s = (a + b + c) / 2
Area: A = √(s(s−a)(s−b)(s−c))
Each term must be positive for a valid triangle
The method works for acute, right, and obtuse triangles
It avoids needing a separate height measurement
Triangle Inequality Validation
Three lengths form a triangle only if each side is smaller than the sum of the other two.
a + b > c
a + c > b
b + c > a
If equality holds, the triangle is degenerate and area is zero
If any condition fails, the calculator returns an error
Finding Altitudes from Area
Once area is known, the height relative to any chosen base follows immediately.
hₐ = 2A / a
hᵦ = 2A / b
h𝒸 = 2A / c
Different bases give different altitudes for the same triangle
These heights are useful in coordinate geometry and construction problems
Worked Examples
Some famous triangles make convenient checks.
3-4-5 → area 6
5-6-7 → area 6√6 ≈ 14.696938
10-10-10 → area ≈ 43.301270
13-14-15 → area 84 exactly
10-10-12 → area 48 exactly
When to Use Heron's Formula
This is the right formula when all three side lengths are known but no altitude is directly measured.
Surveying and land measurement
Triangle checks in construction and framing
Mesh and geometry calculations in graphics
Math education and exam practice
Cross-checking results from trigonometric methods
Historical Background
Heron's formula is named after Hero of Alexandria, though parts of the idea may be older.
Hero documented the formula in Metrica
It is one of the classic formulas of elementary geometry
It remains popular because of its compact form
It connects geometric length data directly to area
It inspired later generalizations such as Brahmagupta's formula
Heronian Triangles
A Heronian triangle has integer side lengths and integer area.
3-4-5 has area 6
5-12-13 has area 30
13-14-15 has area 84
These examples link geometry to number theory
Such triangles are useful classroom examples because calculations simplify nicely
FAQs
What is Heron's formula?
Heron's formula is A = √(s(s-a)(s-b)(s-c)), where s = (a+b+c)/2. It computes triangle area from three side lengths.
Does Heron's formula work for all triangles?
Yes, it works for any valid triangle as long as the three side lengths satisfy the triangle inequality.
What is the semiperimeter?
The semiperimeter is half the perimeter: s = (a+b+c)/2. It is an intermediate value used in Heron's formula.
Why does the calculator check triangle inequality?
If the sum of two sides is not greater than the third side, the lengths cannot form a real triangle, so area is undefined.
Can I find heights from the area?
Yes. Once area is known, the altitude to any side is h = 2A / side. This calculator returns all three altitudes.
What are Heronian triangles?
Heronian triangles have integer side lengths and integer area. Common examples include 3-4-5 and 13-14-15.
Is this the same as a 3-sides triangle area calculator?
Yes, mathematically it uses the same area formula. This version highlights Heron's formula by name and includes the three altitudes explicitly.
What if one side is zero or negative?
A triangle side must be positive. Non-positive values are treated as invalid input.
How precise are the answers?
The calculator computes using floating-point arithmetic and rounds displayed results to 6 decimal places.