Mathematics · Geometry
Heron's Formula Triangle Calculator
Calculate triangle area from all three side lengths.
Inputs and results stay in this browser. Change one value at a time to explore the relationship.
Change an input to reshape this diagram.
Calculation steps
- s=(3+4+5)÷2=6.
- Area=√(6×3×2×1)=6.
Understand Heron's triangle area
One idea, three depths
Choose how deeply to explain Heron's triangle area
Heron's triangle area: Calculate triangle area from all three side lengths.
Age 5Explain it to a 5-year-oldStart with a picture
Imagine using Heron's triangle area to answer this question: calculate triangle area from all three side lengths? Enter Side a, Side b, Side c; the calculator shows Triangle area. For example: Sides 3,4,5 produce area 6. The answer tells you Triangle area.
Age 15Explain it to a 15-year-oldConnect it to the formula
The semiperimeter form obtains area without requiring a measured height. The rule is A=√[s(s−a)(s−b)(s−c)]. Its input values are Side a, Side b, Side c, and the main result is Triangle area. For example: Sides 3,4,5 produce area 6.
CollegeExplain it at college levelState the model precisely
This calculator evaluates the stated heron's triangle area relation over the valid real-number domain stated below. The implemented relation is A=√[s(s−a)(s−b)(s−c)], evaluated from Side a, Side b, Side c to produce Triangle area. The semiperimeter form obtains area without requiring a measured height. The three lengths must satisfy the triangle inequality.
Inputs and valid domain
- Side a must be a finite real number, at least 0.
- Side b must be a finite real number, at least 0.
- Side c must be a finite real number, at least 0.
Important boundary: The three lengths must satisfy the triangle inequality.
The formula
A=√[s(s−a)(s−b)(s−c)]
How the calculator works through it
It substitutes Side a, Side b, Side c into the formula and exposes every numerical step above. The main output is Triangle area, accompanied by Semiperimeter, Perimeter.
Read the result correctly
The Triangle area is the direct answer to “calculate triangle area from all three side lengths.” Read it with the units shown beside the inputs; a sign, angle, percentage or rate changes what the number means.
A worked check
Sides 3,4,5 produce area 6.
Where this model stops being reliable
The three lengths must satisfy the triangle inequality.
Learn it by changing one value
Begin with the worked example, then change one value while keeping the others fixed. Compare the new result and calculation steps to identify which part of the formula changed.
Dictionary terms behind this calculator
Before studying the codeWhat you should know firstUse the calculator immediately, or check the foundations before reading the implementation.
These foundations help you understand why Heron's triangle area works. They never block the calculator, and “optional” means useful context rather than a hidden requirement.
Hard requirements
- Reading formulas and substituting values
Heron's triangle area uses A=√[s(s−a)(s−b)(s−c)]. You need to recognise what each side represents before substituting the stated inputs or rearranging the relationship.
Review this foundation about 4 min
Strong support
- Ratios between measured quantities
Ratios help you check the scale, units and proportional meaning of Heron's triangle area.
Review this foundation about 4 min
Optional enrichment
- Angles and geometric relationships
Angle language provides useful geometric context for extending Heron's triangle area to related shapes and constructions.
Review this foundation about 4 min
Mathematics → algorithm → program
Implement this calculation in code
These are direct reference implementations of the calculator's principal relationship and first output. They run locally and include a small known-answer check where the language supports it.
Algorithm
- Read Side a, Side b, Side c.
- Evaluate the principal relationship: A=√[s(s−a)(s−b)(s−c)].
- Return Triangle area and check the domain conditions described above.
Python
from math import *
def heron_triangle(a, b, c) -> float:
return sqrt((((((a + b) + c) / 2.0) * ((((a + b) + c) / 2.0) - a)) * (((((a + b) + c) / 2.0) - b) * ((((a + b) + c) / 2.0) - c))))
assert abs(heron_triangle(3, 4, 5) - 6) < 1e-6 * max(1.0, abs(6))
C
#include <assert.h>
#include <math.h>
double heron_triangle(double a, double b, double c) {
return sqrt((((((a + b) + c) / 2.0) * ((((a + b) + c) / 2.0) - a)) * (((((a + b) + c) / 2.0) - b) * ((((a + b) + c) / 2.0) - c))));
}
int main(void) {
const double expected = 6;
const double actual = heron_triangle(3, 4, 5);
assert(fabs(actual - expected) < 1e-6 * fmax(1.0, fabs(expected)));
}
C++
#include <cassert>
#include <cmath>
#include <numbers>
double heron_triangle(double a, double b, double c) {
return std::sqrt((((((a + b) + c) / 2.0) * ((((a + b) + c) / 2.0) - a)) * (((((a + b) + c) / 2.0) - b) * ((((a + b) + c) / 2.0) - c))));
}
int main() {
constexpr double expected = 6;
const double actual = heron_triangle(3, 4, 5);
assert(std::fabs(actual - expected) < 1e-6 * std::fmax(1.0, std::fabs(expected)));
}
Linux x86-64 assembly
x86-64 NASM · System V ABI · Linux · SSE2 with libm where required
; double heron_triangle(double a, double b, double c)
; Linux x86-64 NASM · System V ABI · first eight doubles in xmm0–xmm7
global heron_triangle
section .text
heron_triangle:
push rbp
mov rbp, rsp
sub rsp, 208
movsd [rbp-8], xmm0
movsd [rbp-16], xmm1
movsd [rbp-24], xmm2
movsd xmm0, [rbp-8]
addsd xmm0, [rbp-16]
movsd [rbp-72], xmm0
movsd xmm0, [rbp-72]
addsd xmm0, [rbp-24]
movsd [rbp-64], xmm0
mov rax, 0x4000000000000000
movq xmm0, rax
movsd [rbp-80], xmm0
movsd xmm0, [rbp-64]
divsd xmm0, [rbp-80]
movsd [rbp-56], xmm0
movsd xmm0, [rbp-8]
addsd xmm0, [rbp-16]
movsd [rbp-112], xmm0
movsd xmm0, [rbp-112]
addsd xmm0, [rbp-24]
movsd [rbp-104], xmm0
mov rax, 0x4000000000000000
movq xmm0, rax
movsd [rbp-120], xmm0
movsd xmm0, [rbp-104]
divsd xmm0, [rbp-120]
movsd [rbp-96], xmm0
movsd xmm0, [rbp-96]
subsd xmm0, [rbp-8]
movsd [rbp-88], xmm0
movsd xmm0, [rbp-56]
mulsd xmm0, [rbp-88]
movsd [rbp-48], xmm0
movsd xmm0, [rbp-8]
addsd xmm0, [rbp-16]
movsd [rbp-160], xmm0
movsd xmm0, [rbp-160]
addsd xmm0, [rbp-24]
movsd [rbp-152], xmm0
mov rax, 0x4000000000000000
movq xmm0, rax
movsd [rbp-168], xmm0
movsd xmm0, [rbp-152]
divsd xmm0, [rbp-168]
movsd [rbp-144], xmm0
movsd xmm0, [rbp-144]
subsd xmm0, [rbp-16]
movsd [rbp-136], xmm0
movsd xmm0, [rbp-8]
addsd xmm0, [rbp-16]
movsd [rbp-200], xmm0
movsd xmm0, [rbp-200]
addsd xmm0, [rbp-24]
movsd [rbp-192], xmm0
mov rax, 0x4000000000000000
movq xmm0, rax
movsd [rbp-208], xmm0
movsd xmm0, [rbp-192]
divsd xmm0, [rbp-208]
movsd [rbp-184], xmm0
movsd xmm0, [rbp-184]
subsd xmm0, [rbp-24]
movsd [rbp-176], xmm0
movsd xmm0, [rbp-136]
mulsd xmm0, [rbp-176]
movsd [rbp-128], xmm0
movsd xmm0, [rbp-48]
mulsd xmm0, [rbp-128]
movsd [rbp-40], xmm0
sqrtsd xmm0, [rbp-40]
movsd [rbp-32], xmm0
movsd xmm0, [rbp-32]
leave
ret
MATLAB
function result = heron_triangle(a, b, c)
result = sqrt((((((a + b) + c) / 2.0) * ((((a + b) + c) / 2.0) - a)) * (((((a + b) + c) / 2.0) - b) * ((((a + b) + c) / 2.0) - c))));
end
Wolfram Language
ClearAll[mwCalculate];
mwCalculate[a_, b_, c_] := Sqrt[(((((a + b) + c) / 2.0) * ((((a + b) + c) / 2.0) - a)) * (((((a + b) + c) / 2.0) - b) * ((((a + b) + c) / 2.0) - c)))];
Continue in mathematical software
The downloaded file includes your current inputs and first calculated result. It is created locally.
Floating-point answers can differ slightly by language, compiler and processor. Compare within a suitable tolerance rather than assuming every decimal representation will be identical.
Supporting sourcesAcademic referencesPrimary standards, textbooks and complete citations
Standards, reading and academic references
Use the calculator as the worked interaction, then consult the primary standards and academic textbooks listed below. MW SysArc links to the original sources; the explanation on this page is original and does not reproduce them.
Algebra and Trigonometry 2e
Read the related free OpenStax mathematics chaptersCite this book
- APA 7
- Abramson, J. (2021). Algebra and trigonometry 2e. OpenStax. https://openstax.org/books/algebra-and-trigonometry-2e/pages/1-introduction-to-prerequisites
- MLA 9
- Abramson, Jay. Algebra and Trigonometry 2e. OpenStax, 2021, https://openstax.org/books/algebra-and-trigonometry-2e/pages/1-introduction-to-prerequisites.
- Chicago author-date
- Abramson, Jay. 2021. Algebra and Trigonometry 2e. Houston, TX: OpenStax. https://openstax.org/books/algebra-and-trigonometry-2e/pages/1-introduction-to-prerequisites.
OpenStax entries are free to read online. Follow the licence shown on each linked source before redistributing or adapting its content.
Reuse the page responsiblyCite this pageAPA, MLA, Chicago, Harvard, BibTeX and RIS
These formats cite this calculator page itself. They are separate from the academic references above, which support the mathematical method and terminology.
APA 7
MW SysArc. (2026, July 21). Heron's Formula Triangle Calculator. MW SysArc Tools. https://math.mwsysarc.com/geometry/herons-formula-triangle
MLA 9
MW SysArc. “Heron's Formula Triangle Calculator.” MW SysArc Tools, 21 July 2026, https://math.mwsysarc.com/geometry/herons-formula-triangle. Accessed 31 Aug. 2026.
Chicago 17
MW SysArc. “Heron's Formula Triangle Calculator.” MW SysArc Tools. Published July 21, 2026. Accessed August 31, 2026. https://math.mwsysarc.com/geometry/herons-formula-triangle.
Harvard
MW SysArc (2026) ‘Heron's Formula Triangle Calculator’, MW SysArc Tools. Published 21 July 2026. Available at: https://math.mwsysarc.com/geometry/herons-formula-triangle (Accessed: 31 August 2026).
BibTeX and RIS records
BibTeX
@misc{mwsysarc_heron_triangle_2026,
author = {{MW SysArc}},
title = {Heron's Formula Triangle Calculator},
howpublished = {MW SysArc Tools},
year = {2026},
url = {https://math.mwsysarc.com/geometry/herons-formula-triangle},
note = {Published July 21, 2026; accessed August 31, 2026}
}RIS
TY - ELEC
AU - MW SysArc
TI - Heron's Formula Triangle Calculator
T2 - MW SysArc Tools
PY - 2026
DA - 2026-07-21
Y2 - 2026-08-31
UR - https://math.mwsysarc.com/geometry/herons-formula-triangle
N1 - Published July 21, 2026
ER -Clear answers
Frequently asked questions
What does the Heron's triangle area do?
Calculate triangle area from all three side lengths.
How does the Heron's triangle area work?
The calculator applies A=√[s(s−a)(s−b)(s−c)]. The semiperimeter form obtains area without requiring a measured height.
What can I learn from the Heron's triangle area?
It connects the mathematical rule to your chosen numbers and shows each calculation step. Change one input at a time to see how the result responds.
Does MW SysArc receive or store what I enter?
No. The calculation runs locally in your browser. MW SysArc does not receive or store your calculation inputs.
How should I use the result?
Use the steps to understand the method, then verify important school or professional work using the notation and rounding rules required in your setting.
Last reviewed . Calculations tested .