Volume bepalen met behulp van doorsnedeoppervlakte

For solids where the cross-sectional areas are known, the volume is found by integrating these areas along an axis that is perpendicular to the slices.

Consider a tetrahedron with height h and a base that is an equilateral triangle with side a.

The origin is placed at the top vertex, with the y-axis extending along the height.

Cross-sections taken perpendicular to this axis form smaller equilateral triangles that increase in size from the vertex to the base.

With one base side parallel to the x-axis, the side length of each small equilateral triangle at height y is found using similar triangles. It scales as y over h times a.

The area of each equilateral triangle equals root 3 over 4 times the square of its side. These areas show the changing cross-sections along the height.

The volume of the tetrahedron is found by integrating these areas from the vertex to the base. The result is proportional to the product of the height and the square of the base side.

Surveyors use this method to get the volume of an irregular gravel pile by measuring changing cross-sections and integrating them for accurate material estimates.