Encyclopedia

Projected area

Two-dimensional area measurement of a three-dimensional object projected onto a plane

Wikipedia contributorsAdapted for Fathomly · View authors

1 min read
Source updated January 27, 2026

Projected area is the area measurement of a three-dimensional object by projecting its silhouette 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: $A_{\text{projected}}=\int _{A}\cos {\beta }\,dA$ where A is the original area, and $\beta$ 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.

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

Watch videos about Projected areaExplainers and documentaries on YouTube (opens in a new tab)

Sources and credits

This article is adapted from the Wikipedia article “Projected area”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

Images, from Wikimedia Commons:

Projected area-hardness test.svg by Wizard191, CC BY-SA 3.0

Fathomly is not affiliated with or endorsed by the Wikimedia Foundation. Spotted a problem? Tell us.

Projected area

Sources and credits

Related topics

Cross section (geometry)

Surface area