Question

How can you use proportions and similar triangles to indirectly measure large objects, such as heights of buildings and mountains?

Answer 1

Form a virtual triangle using the object's height and its shadow's length as bases.

Form another virtual triangle using another object (a much smaller, measurable object. A tree, a pole, or a person for example) and its shadow. The other object should be in the same vicinity as the object being measured so that the shadows would form the same angles.

You now have your similar triangles.
Similar triangles have proportional dimensions.

This means we can equate the ratios of the bases.

Let `L` be the large object being measured.
`L_H` is `L`'s height.
`L_S` is `L`'s shadow's length

Let `S` be the object used for comparison.
`S_H` is `S`'s height
`S_S` is `S`'s shadow's length

Get the length/height of `L_S`, `S_H`, and `S_S`.
Then, for `L_H`, we have

`L_H/L_S = S_H/S_S`

`=> L_H = S_H/S_S * L_S`