Answer to Question #84771 in Optics for hamidouche massine

Question

While outside watching the projection of the eclipse with their very nice pinhole projector, the students noticed images of the eclipse on the ground under the tree nearby The images were not only larger than their pinhole projection, but there were many images of the sun to see all at once.
One student took out a ruler and measured the image of the sun on the ground to have a diameter of 6 cm. Approximately how tall was the tree? Enter your answer in meters.

Answer 1

The images of the sun on the ground were formed by solar light rays passing through small holes between the tree leaves, the whole situation thus forming natural pinhole projectors with the ground playing the role of screen. The distance h between the ground and the highest of such holes approximately gives the height of the tree. This distance can be found from the relation

d = h sin θ , (1)

where d = 6 cm = 0.06 m is the diameter of the largest image of the sun on the ground, and θ is the angular diameter, measured in radians, of the same image viewed from the position of the hole. Observe that θ is also the angular diameter of the sun viewed from the earth, which is approximately equal to 0.5°, or 0.5×2π/360 radians ≈ 0.0087 radians. From relation (1), we have h = d/sin θ ≈ d/θ, where we estimated sin θ by θ in view of the smallness of θ relative to unity. Substituting the numbers, we obtain the estimate of the height of the tree: h ≈ 0.06 m / 0.0087 ≈ 6.9 m.