Triangle Area Calculator

Formula

A = (b × h) / 2

Area of a triangle: three formulas to know

The triangle is the simplest polygon, and which area formula you reach for depends on what you already know. The classic one takes a base and its matching height: A = b·h / 2, half of the rectangle that would box it in. If all you have are the three sides, Heron's formula handles it: A = √(s(s-a)(s-b)(s-c)), with s = (a+b+c)/2 as the semi-perimeter. And when you know two sides and the angle between them, there's the trigonometric form: A = (1/2)·a·b·sin(C).

Try Heron on the Egyptian 3-4-5 right triangle. First the semi-perimeter, s = (3+4+5)/2 = 6, then A = √(6·3·2·1) = √36 = 6. Now check it the other way, using the legs 3 and 4 as base and height: 3·4/2 = 6. Same answer, which is what you'd expect.

Where the triangle shows up in practice

Think of geodetic triangulation, the method that mapped whole continents before GPS, or GPS positioning itself, which trilaterates your distance from satellites. Computer graphics belongs here too: every 3D model you see on screen is a mesh of triangles, since the triangle is the simplest polygon a GPU knows how to rasterize. In Brazilian land surveying, INCRA wants rural plots reported in hectares, where 1 ha = 10,000 m², and big plots are usually broken into triangles whose areas get added up one by one.

FAQ

Which formula should I use? Go by your measurements. Base and height point you to b·h/2. Three sides only, as you often get with land plots, calls for Heron. Two sides and the angle between them, common in surveying, fits the trigonometric form.

Does the height have to fall inside the triangle? No. In an obtuse triangle, the height drawn to a given base can land outside the figure, on the extension of that side. The formula still gives the right area.

How do I convert m² to hectares? Divide by 10,000. So 25,000 m² works out to 2.5 ha. Going from hectares to alqueire paulista, multiply by about 0.413.