Calculadora de área do triângulo

The most common way to find a triangle's area is base × height ÷ 2, where height is measured perpendicular to the base. This works for any triangle as long as you know a base and the corresponding height.

When you only know the three side lengths (and not the height), Heron's formula solves it instead: first find the semi-perimeter s = (a + b + c) / 2, then area = √(s(s−a)(s−b)(s−c)). If the three lengths cannot form a valid triangle (one side is too long relative to the other two), the area comes out to 0.

Example: a triangle with base 10 and height 4 has area 10 × 4 ÷ 2 = 20. A 3-4-5 right triangle, using Heron's formula with s = 6, gives area = √(6×3×2×1) = √36 = 6 — matching the simpler ½ × 3 × 4 = 6 you'd get treating the legs as base and height.

Heron's formula works for any triangle when you know three sides — no height needed. It fails (area 0) when sides violate the triangle inequality.

Equilateral triangle with side s: area = (√3/4)s² — faster than Heron when all sides match.

Worked example in base-height mode: base 10, height 4 → area 20. Same 3-4-5 triangle in Heron mode (sides 3,4,5) also gives area 6 using half the legs as base and height.

Coordinate geometry shoelace formula is an alternative when vertices are on a grid — base×height mode and Heron mode should agree on area for the same triangle when inputs are consistent. Shoelace on grid vertices should match base×height — inconsistent inputs between modes signal typo in side or height entry. Cross-check shoelace versus base×height when triangle vertices sit on a coordinate grid.

Oblique triangles need height perpendicular to chosen base — slant height of a roof face is not the height for area unless you project vertically to the base line. Roof slant height is not altitude to base — project vertical height perpendicular to the base line for area, not rafter length alone. Draw the altitude perpendicular to your chosen base, not the slant rafter length alone.

Scalene 5-6-7 triangle in Heron mode: s=9, area=√(9×4×3×2)=√216≈14.7 sq units — compare to base-height if you drop altitude from longest side. Heron on 5-6-7: s=9, area √216 ≈ 14.7 — compare to base-height if you drop altitude from the longest side instead.