Triangle calculator
Triangle area from three sides or base and height; perimeter and classification require all three sides.
Fill in the fields and the result will appear here automatically.
Works a triangle two ways: from three sides by Heron’s formula, or from a base and its height as half their product. Three sides are checked against the triangle inequality first — if any two do not exceed the third, the figure does not exist, and the calculator says so instead of returning a zero that reads like an answer. It also reports the kind of triangle: right, acute or obtuse.
How it works
Formula and logic
First p = (a+b+c)/2, then S = √(p(p−a)(p−b)(p−c)). The four factors are p and three differences. The calculation uses the equivalent expression S = √[(a+b+c)(−a+b+c)(a−b+c)(a+b−c)/16], retaining small differences before rounding. From base a and perpendicular height h: S = ah/2. Perimeter a+b+c and classification by squared sides require all three sides.
Example
A triangle with sides of 3, 4 and 5 m is right-angled: its area is 6 m² and its perimeter 12 m.
Fields and units
- Length unit — list option
- What is known — list option
- Side a — cm
- Side b — cm
- Side c — cm
- Base — cm
- Height — cm
How to use
- — Choose three sides, or a base and its corresponding perpendicular height.
- — Enter positive lengths in one unit.
- — In sides mode any two sides must strictly exceed the third; perimeter and angle classification are available only in that mode.
- — Near a flat triangle a small measurement error can strongly change the area.
Method and limitations
- Calculation method
- Formula and logic
- Data or methodology source
- OpenStax: area, perimeter and squared units W. Kahan: numerical errors of Heron’s formula for needle-like triangles — lecture notes
- Limitation
- Only planar triangles of positive area. Collinear sides are excluded by this calculation’s scope. Classification compares the entered numbers without a measurement tolerance; it does not certify a real object’s angle. Results are rounded.
FAQ
Why are some sets of sides rejected?
Positive area requires any two sides to sum to more than the third. With sides 1, 2 and 3 the points lie on a line: a degenerate zero-area case excluded by this calculator’s scope.
What is Heron’s formula?
First p = (a+b+c)/2, then S = √(p(p−a)(p−b)(p−c)). The four factors are p and three differences. The calculation uses the equivalent expression S = √[(a+b+c)(−a+b+c)(a−b+c)(a+b−c)/16], retaining small differences before rounding.
How is the kind of triangle decided?
By comparing the square of the longest side with the sum of the squares of the other two: equal means right-angled, smaller means acute, larger means obtuse.
Does the height have to belong to the entered base?
Yes. The height must be dropped onto the base you entered, otherwise half their product is not the area of this triangle.
Can base and height determine a triangle’s perimeter?
No. Triangles with the same base and height have the same area but may have different other sides. Base 6 and height 4 give area 12; the symmetric triangle has two sides of 5, while moving its vertex changes those sides.