The surface of the right-angled triangle

The turquoise right-angled triangle and the white right-angled triangle share 3 angles and 1 side. They are congruent and have equal surface.

The surface of the rectangle

= a b   (a and b are real numbers)

= 2 * the surface of the turquoise right-angled triangle.

The surface of the turquoise right-angled rectangle = a b / 2

The surface of the triangle

Approach the turquoise right-angled triangle as a stack of n bars of thickness h/n (first black-and-white figure).

The surface of the right-angled triangle is (nearly) the sum of the surface of the bars.

Approach every other triangle of same base b and height h by shifting the bars (second and last black-and-white figure).

The total surface of the bars doesn't change when bars are shifted.

For n approaching infinity, the bars approach line segments and the total surface of the bars approaches the surface of the triangle.

The horizontal width at same height and parallel to the base b is precisely the same line segment in every triangle.

Every triangle of same base and height is a stack of precisely the same line segments.

The surfaces of all triangles of same base and same height are the same.

Surface of the triangle with height h and base b = hb / 2