# Projected area

Example of a projected area from a hardness indentation.

Projected area is two-dimensional area measurement of a three-dimensional object by projecting its shape on to an arbitrary plane. This is often used in mechanical engineering and architectural engineering related fields, specifically hardness testing, axial stress, wind pressures, and terminal velocity.

The geometrical definition of a projected area is: "the rectilinear parallel projection of a surface of any shape onto a plane". This translates into the equation:

${\displaystyle A_{projected}=\int _{A}\cos {\beta }\,dA}$

where A is the original area, and ${\displaystyle \beta \,}$ is the angle between the normal to the surface A and the normal to the arbitrary plane onto which we project. For basic shapes the results are listed in the table below.[1]

Projected area for basic shapes[1]
Shape Area Projected area
Flat rectangle ${\displaystyle A=L\times W}$ ${\displaystyle A_{proj}=L\times W\cos {\beta }}$
Circular disc ${\displaystyle A=\pi r^{2}}$ ${\displaystyle A_{proj}=\pi r^{2}\cos {\beta }}$
Sphere ${\displaystyle A=4\pi r^{2}}$ ${\displaystyle A_{proj}={\frac {A}{4}}=\pi r^{2}}$

## References

1. ^ a b [1]Palmer, James M. (1999-07-08), Radiometry and photometry FAQ (PDF), retrieved 2011-04-02.