Napoleon Triangle Area Calculator
Compute outer/inner Napoleon triangle areas from actual noncollinear planar vertices and erected equilateral-triangle centroids; inner construction may degenerate.
Description
Napoleon triangles are formed from centers of equilateral triangles erected on a triangle's sides.
Napoleon triangles are formed from centers of equilateral triangles erected on a triangle's sides. The outer and inner constructions have different areas, and the inner one may degenerate.1
Compute outer/inner Napoleon triangle areas from actual noncollinear planar vertices and erected equilateral-triangle centroids; inner construction may degenerate.
When to use Napoleon Triangle Area Calculator
Use this tool to compute outer/inner Napoleon triangle areas from actual noncollinear planar vertices and erected equilateral-triangle centroids; inner construction may degenerate.
Choose the integer, sequence, coefficients or function parameters for your calculation. Follow the stated ordering, domain and size limits.
How to use Napoleon Triangle Area Calculator and understand its results
- Choose the integer, sequence, coefficients or function parameters for your calculation. Follow the stated ordering, domain and size limits.
- Enter vertices using the field table and calculation rules, then run the tool.
- Read originalTriangleArea, outerNapoleonTriangleArea, innerNapoleonTriangleArea. The explanation below defines the units, signs, ordering and any flags or placeholder values.
- If the tool reports an input issue, correct the values and run it again. Apply the stated assumptions when interpreting a successfully returned result.
| Input field | Meaning | Value type; see rules below | Requirement |
|---|---|---|---|
| vertices | Vertices | list of list of number | Required |
Enter three well-conditioned noncollinear planar vertices. Calculate original area and the common squared-side contribution; return outer area and nonnegative inner area, clamping tiny negative roundoff to zero.
base = sum of squared side lengths/(8sqrt(3)); outer/inner area = base ± original area/2 Units, indexing, boundary conditions and exceptional cases are defined in the calculation rules above.
| Output field | Meaning | Value type |
|---|---|---|
| originalTriangleArea | Original triangle area | number |
| outerNapoleonTriangleArea | Outer napoleon triangle area | number |
| innerNapoleonTriangleArea | Inner napoleon triangle area | number |
{
"input": {
"vertices": [
[
0,
0
],
[
4,
0
],
[
0,
3
]
]
},
"output": {
"originalTriangleArea": 6,
"outerNapoleonTriangleArea": 6.608439182435161,
"innerNapoleonTriangleArea": 0.6084391824351614
}
}Limitations and assumptions
Results use the specified outer/inner equilateral construction. No coordinates of the erected triangles are returned.
Numeric inputs and returned numeric values must be finite; an unrepresentable returned value produces an error. Matrices are limited to 128 rows and 128 entries per row; other numeric lists are limited to 100,000 entries. Missing required fields, extra fields, wrong value types and violations of the calculation rules are input errors.
Alternative or Complementary methods
Use Nine Point Circle Calculator: Construct nine-point circle and actual triangle side midpoints; center uses circumcenter/orthocenter and radius is half circumradius, not a generic circle from radius. For a complementary task, use Euler Line Calculator: Construct circumcenter, centroid, orthocenter and Euler-line direction from planar vertices; equilateral coincident centers leave no unique Euler line.
References
-
Napoleon Triangle Area Calculator — tool.io
Similar or alternative tools
- Nine Point Circle Calculator
Construct nine-point circle and actual triangle side midpoints; center uses circumcenter/orthocenter and radius is half circumradius, not a generic circle from radius.
- Euler Line Calculator
Construct circumcenter, centroid, orthocenter and Euler-line direction from planar vertices; equilateral coincident centers leave no unique Euler line.