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Measuring heights with angles and shadows

Stand 30 m from a tree, look up 40° to its top, and the tree is about 25.2 m taller than your eyes. No ladder, no tape up the trunk.

Angles of elevation and depression

OpenStax defines the angle of elevation as the angle between the horizontal and your line of sight when you look up at something. Looking down from a height, the angle between the horizontal and the line of sight is the angle of depression. Both are measured from the horizontal, never from the vertical. Surveyors measure them with an instrument on a tripod; a protractor with a weighted string, or a phone app, works for a school project.

α is an angle of elevation, measured up from the horizontal at the eye. β is an angle of depression, measured down from the horizontal at the top. Both dashed lines are horizontal.

Height from one angle

Walk a measured distance d from the foot of a tree and measure the angle of elevation θ to its top. The horizontal line from your eye, the vertical trunk and your line of sight form a right triangle. The distance d is the side adjacent to θ, the height above your eye is opposite, so the tangent links them: height above eye = d tan θ. OpenStax works this with a tree 30 feet away at 57°, giving about 46.2 feet. Do not forget your own eye height. At 30 m and 40°, with eyes 1.5 m above the ground, the tree is 1.5 + 30 tan 40° ≈ 1.5 + 25.17 ≈ 26.67 m tall.

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A 24 m tower. Drag, or move the angle slider, until the line of sight just meets the top; the estimate 1.5 + d tan θ then matches the tower. Switch Method to Shadows for the Thales way.

Looking down

From a window 12 m above the street you see a parked car at an angle of depression of 25°. The horizontal line at the window and the street are parallel, and your line of sight crosses both, so the angle of depression equals the angle of elevation from the car up to you: they are alternate interior angles. The car is therefore 12 / tan 25° ≈ 25.73 m from the foot of the building. With two angles from one window you can measure a whole tower across the street. If it is 50 m away, the elevation to its top is 30° and the depression to its foot is 20°, the tower is 50 tan 30° + 50 tan 20° ≈ 28.87 + 18.20 ≈ 47.07 m tall.

Shadows

BackgroundMacTutor collects the old accounts of how Thales of Miletus measured the height of an Egyptian pyramid. Diogenes Laertius, writing in the second century AD, quotes Hieronymus, a pupil of Aristotle: Thales measured the pyramids by their shadow at the moment when our shadows are equal to our own height. Plutarch tells a version closer to geometry. Thales set up a stick at the tip of the pyramid's shadow and, with the two triangles made by the sun's rays, showed that the pyramid is to the stick as the pyramid's shadow is to the stick's shadow. MacTutor adds that these stories were written centuries after Thales lived.

Both versions work because the sun is so far away that its rays arrive parallel. Every upright object then has the same ratio of height to shadow, which is the tangent of the sun's elevation. If a 1 m stick casts a 1.6 m shadow and a tower casts a 40 m shadow at the same moment, the tower is 40 × 1/1.6 = 25 m tall. At the special moment when the sun is 45° high, tan 45° = 1 and every shadow is exactly as long as its object: the first version of the story.

  1. You stand 30 m from a tree with eyes 1.5 m high and see its top at 40°. Height of the tree to 2 decimal places?
  2. OpenStax: a tree 30 ft away, angle of elevation 57° measured from the ground. Height to 1 decimal place?
  3. In the lab: tower 24 m, d = 30 m, eyes at 1.5 m. Which whole-degree angle hits the top most closely?
  4. From a window 12 m up, a car is seen at a depression angle of 25°. Its distance from the building to 2 decimal places?
  5. Why does the angle of depression from the window equal the angle of elevation from the car?
    • they are alternate interior angles between parallel lines
    • they are vertical angles
    • they are complementary
    • the building is tall
  6. Tower 50 m away: elevation to the top 30°, depression to the foot 20°. Height to 2 decimal places?
  7. A 1 m stick casts a 1.6 m shadow; a tower casts a 40 m shadow. Height of the tower?
  8. Same moment. Sun elevation to 1 decimal place?
  9. At what sun elevation is every shadow as long as its object?
  10. An angle of elevation is measured from:
    • the horizontal
    • the vertical
    • the ground at the object
    • the line of sight
  11. As the sun gets lower in the sky, shadows:
    • get longer
    • get shorter
    • stay the same
    • disappear
Key points
  • Elevation and depression are measured from the horizontal.
  • Height above the eye = d tan θ; add the eye height.
  • Depression from above = elevation from below (alternate interior angles).
  • Shadows: height / shadow is the same for every object at one moment; at 45° they are equal.