Mathematics · Geometry
Calculate triangle area three ways
Pick the data you have: a base and its perpendicular height, three side lengths, or two sides and the included angle. A collapsed (zero-area) triangle is rejected.
Calculate
How the calculator works
Base–height uses only base and height. Heron uses a, b, c: first the semiperimeter s, then the square root; triangle inequalities are enforced. SAS uses a, b, and the angle in degrees converted to radians for sin. Fields that do not belong to the chosen mode are ignored.
Formula and method
A = (base × hauteur) / 2
Height must be perpendicular to that base. Base 10 and height 4 give area 20.
A = √[s(s − a)(s − b)(s − c)] with s = (a + b + c) / 2
Heron. Sides 3, 4, 5: s = 6, A = √(6×3×2×1) = √36 = 6.
A = (1/2) a b sin C
SAS. Sides 3 and 4 with included 90°: (1/2)×3×4×1 = 6.
Degrees, not gradians. A flat angle (0° or 180°) makes sin 0 and is rejected, as are side lengths that cannot form a triangle. Heron’s formula can be numerically delicate for needle-thin triangles; impossible inputs are caught before a NaN area. Pythagoras is not required, but a 3-4-5 right triangle is a useful check across modes.
Worked example
Base 10, height 4
mode base–height, base = 10, height = 4
- A = (10 × 4) / 2 = 20.
Area 20 square units.
Heron 3-4-5
mode Heron, a = 3, b = 4, c = 5
- s = 6.
- A = √(6×3×2×1) = 6.
Area 6, matching (3×4)/2 for a right triangle.
Input notes
- Method
- Determines which fields are read.
- Base
- Any side you treat as base, with the matching perpendicular height.
- Height
- Altitude to that base, not a slanted side.
- Side a
- Heron or SAS side.
- Side b
- Heron or SAS side.
- Side c
- Heron only.
- Included angle (degrees)
- Included angle between a and b in SAS, in degrees.
Assumptions and limits
Assumptions
- Euclidean plane triangle.
- SAS angle is the included angle, not a remote angle (SSA is not offered).
Limits
- SSA (two sides and a non-included angle) is ambiguous and not implemented.
- No 3-D surface area.
How to read the result
Area 6 for 3-4-5 is also (leg × leg)/2. If Heron and base–height disagree, the height you typed is not the altitude to that base, or a side was mistyped. For a right triangle’s hypotenuse, use Pythagoras; area still needs a height or two legs.
Common mistakes
Using a slanted side as “height.”
Height is perpendicular to the chosen base.
Entering SAS degrees when you meant radians, or 90 for a 90-radian angle.
The field is degrees. A right angle is 90.
Related calculations
- Circle areaOne length is enough: the radius, or the diameter (halved). You get area πr², circumference 2πr, and both r and d as a check.
- Pythagorean theoremTwo modes: √(a² + b²) for the hypotenuse, or √(c² − a²) for the missing leg. The triangle must be right-angled; a “leg” longer than the hypotenuse is rejected.
Frequently asked questions
Why was my Heron input rejected?
The three lengths fail the triangle inequality, or a side is not positive.
Each side must be shorter than the sum of the other two. 1, 2, 3 is a degenerate straight line (area 0) and is refused.
Can I use SAS with an angle that is not included?
No. That is SSA, which can have 0, 1, or 2 triangles.
This page only implements the included-angle case, where the area formula is (1/2)ab sin C.
Is 3-4-5 a special case?
It is a right triangle, so several modes should agree.
Base 3 height 4, Heron 3-4-5, and SAS 3, 4, 90° all give area 6. That is a useful self-check, not a requirement.
Author and update
Written by Rédaction HexaCalc (editorial team). Content last updated: August 25, 2026. No third-party medical or financial review is claimed.
- Héron d’Alexandrie — Metrica (formule de Héron) — Tradition mathématique grecque
Missing a side of a right triangle?
Area needs enough data to fix the shape. If you only lack the hypotenuse, solve a² + b² = c² first.
Use the Pythagorean theoremCategory: Mathematics