Estimating Tree Height Using Natural Pinhole Cameras

August 15, 2016

EllipticalSplash20160810_092642 (1)

Photographer: John Adam
Summary AuthorJohn Adam

Have you ever noticed the almost elliptical splashes of sunlight on the ground or sidewalk as you walk under a tree on a sunny day? Suppose the ellipse has a major axis of b cm and a minor axis of a cm. By using simple proportion and some simplifying assumptions (such as the angle subtended by the light patch at the pinhole is small) it can be shown that the height of any particular pinhole producing the patch is approximately

108a2/b cm or 1.08a2/b m

This is also a reasonable estimate for the height of the tree if the pinhole is near the top of the canopy. But where does the factor of 108 come from? The angular diameter of the Sun as seen at the surface of the Earth is about 0.53 degrees of arc or 1/108 radians. This is used in the geometrical derivation of the above formula.

On another occasion while walking home along a path lined with crape myrtle trees (elliptical splashes shown above), I measured values (in inches) for a and b of two and three in (5 cm and 7.6 cm) respectively, so the height of the tree pinhole was about 12 ft (3.7 m), which looked about right! Photo taken near Old Dominion University in Norfolk, Virginia, on August 10, 2016.

Photo Details: Camera Model: SAMSUNG-SGH-I337; Focal Length: 4.2mm (35mm equivalent: 31mm); Aperture: ƒ/2.2; Exposure Time: 0.025 s (1/40); ISO equiv: 50.