# Projected Areas of Cylinder and Cone

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This Demonstration calculates the projected area of a cylinder topped by a cone. This represents the frontal area for fluid flow incident on the cylinder.

Contributed by: Aaron T. Becker and Yitong Lu (January 2021)

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

Consider a cylinder of radius and height rotated with respect to the flow direction of a fluid by about an axis parallel to the base. The frontal area of the cylinder is the area perpendicular to the flow direction. If this shape is projected onto the 2D plane, the resulting 2D area is

.

If , this is just the rectangle . When , the area is that of the circular end cap .

A right cone with radius and height is more complicated. If , the projected area is just the triangle . When , the area is that of the circular base . If , the cone is rotated so far that the tip of the cone is contained within the projection of the base, so the area is , which is the circular base area multiplied by .

Otherwise, the area is composed of the union of the ellipse projected by the base of the cone and a triangle from the apex of the cone to points tangent to this ellipse.

These tangent points can be computed by solving for the line that passes through the projection of the apex of the cone and touches the ellipse at just one point. The tangent points (shown as black points) are

.

The resulting area of the cone is

.

The frontal area is the reference area often used for calculating the coefficient of drag, and is typically defined as the area of the orthographic projection of the object on a plane perpendicular to the direction of motion.

## Permanent Citation