Shadows

Shadows

Change of light conditions in different seasons. Measuring height using shadows.

Nature

Keywords

shadow, light condition, measuring height, winter solstice, summer solstice, path of the Sun, season

Related items

Scenes

  • December 21 on the northern hemisphere June 21 on the southern hemisphere
  • June 21 on the northern hemisphere December 21 on the southern hemisphere
  • March 21 and September 22

  • 10 m (32.81 ft)
  • 8.3 m (27.23 ft)
  • 67.2 m (220.5 ft)
  • 81 m (265.7 ft)

We know the length of both legs of the small right-angled triangle, and one of the legs of the large right-angled triangle.
In this example, the ratio of the two legs is: 10 m (32.81 ft) : 8.3 m (27.23 ft)

The length of one leg of the large right-angled triangle (the length of the shadow) is known, while the other (the height of the tower) is unknown.
In this example, their ratio is: x m: 67.2 m (220.5 ft)

The ratio of the corresponding legs of similar triangles is equal:
10 m (32.81 ft) : 8.3 m (27.23 ft) = x m : 67.2 m (220.5 ft)
Therefore
x = (10 m : 8.3 m or 32.81 ft : 27.23 ft) ∙ 67.2 m (220.5 ft) ≈ 81 m (265.7 ft).
The tower is approximately 81 m (265.7 ft) tall.

  • 10 m (32.81 ft)
  • 8.3 m (27.23 ft)
  • 67.2 m (220.5 ft)
  • 81 m (265.7 ft)

Narration

The winter solstice marks the longest night of the year. After this, days become longer until the summer solstice, which marks the longest day of the year. In the Northern Hemisphere, the winter solstice usually occurs on December 21. In the Southern Hemisphere, it is the summer solstice that occurs on this date.

In the Northern Hemisphere, the summer solstice usually occurs on June 21, which is the day the winter solstice occurs in the Southern Hemisphere.

This animation demonstrates shadows in countries located in Central Europe or at similar latitudes. As you can see, shadows are longer at the time of the winter solstice than at the summer solstice, since the apparent path of the Sun is lower in winter than in summer. In the Southern Hemisphere, it is the other way round.

The height of a building can easily be ascertained with shadows. To do this, we need an object of known height, then we measure the length of its shadow. In this example we know that the height of the pole is 10 m (32.81 ft). We measure the length of its shadow, then we measure the length of the tower's shadow.

We can now calculate the height of the tower using our knowledge of similar triangles.

We know the lengths of both legs of the small triangle. We also know the length of one leg of the large triangle. Based on these figures, we can calculate the length of the other leg, that is, the height of the tower.

Since the two triangles are similar, the ratio of the lengths of the corresponding legs is equal. That is, the ratio of the pole height to the length of the shadow of the pole is equal to the ratio of the tower height to the length of the tower's shadow. Now let's solve the equation to find the height of the tower.

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