In their book Color and Light in Nature,1 authors Dave Lynch and Bill Livingston make the following enigmatic statement:

“Cast a shadow with your hand onto a piece of paper. See how the sharpness of the shadow’s edge diminishes as you raise your hand above the surface. That fuzzy outer edge of the shadow is the penumbra. Its width divided by its distance from your hand is always a constant fraction of about 1/112.”

Question 1: Can you explain this statement using elementary geometry (e.g., similar triangles)? (Using the approximation that the angular diameter of the Sun at your eye is ~0.5° of arc will result in a slightly different fraction from the above.)

Solution to Question 1: From Fig. 1, $puL=dD−(L+x)≈dD$. But $0.5∘=π360≈1120rad$.

A closer approximation for 0.5° is about 1/115 rad, and since the average angular diameter of the Sun on Earth is 0.53° of arc, the corresponding fraction is about 1/108, so the range includes Lynch and Livingston’s result!

Question 2: Use the same geometry to explain why the shadow of a tall object (a flagpole, for example) becomes less sharply defined as you gaze away from the shadow base.

Solution to Question 2: Mentally move the screen to P, the tip of the shadow. Then the above geometry holds true with the left-hand side being $puL+x$.