A triangle in a rectangle always fills half its area
Triangles on one base between parallels enclose equal area, whatever the apex
These figures state the classic area law for triangles. Leonardo shows that a triangle inscribed in a rectangle (a column) with its base on the rectangle's base and its apex touching the top occupies exactly half the rectangle, so the triangle equals the remaining area. He generalises this to triangles drawn between two parallels on one common base: however far the apex slides sideways, they all enclose the same area. Labels a, b, c, d mark the column and the divided rectangles.
On this page
The largest triangle equals the rest of the column
For any rectangular column of equal width, the largest triangle with equal base angles drawn inside it has an area equal to the remainder of the column. The bases c and d span the width below and a and b span it above; the four equal triangles split so that the triangle holds two and the remainder holds two.
Any triangle occupies half of its square
In any square with equal angles, a triangle whose whole base lies on the square's base and whose tip touches the upper side always occupies half the area of the square. This holds whether the square and triangle are equilateral or not.
Triangles on one base between parallels are equal
Every triangle set between two parallels, reaching from the lower to the upper line and founded on one same base, necessarily contains the same quantity of area. A long rectangle shows one upright triangle and one strongly elongated triangle sharing the base but with far-apart apexes.
A divided rectangle (a, b, c, d)
An irregularly subdivided rectangle marks four intersection points labelled d, b, a, c. A similar figure is repeated alongside it.