Projected area

Projected area is the 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, especially for 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: $${\displaystyle A_{\text{projected}}=\int _{A}\cos {\beta }\,dA}$$ where A is the original area, and ${\displaystyle \beta }$ is the angle between the normal to the local plane and the line of sight to the surface A. For basic shapes the results are listed in the table below.^[1]

Projected area for basic shapes^[1]

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