Math

Since we do not know the exact date or time we assume that the shadow falls due north. (Pisa is north of the Tropic of Cancer so the sun will appear in the southern sky around noon.)

The tower leans `5.5^@` from the vertical towards the south. The length of the shadow to the north is 84.02m and the angle of elevation is `32.0^@` . Draw a triangle with side 84.02 such that this side is included between the angles measuring `32^@` and `95.5^@` . (We use `95.5^@` since the tower if vertical would form a 90 degree angle; but it leans `5.5^@` away from the shadow.) The third angle of the triangle, opposite the side of known length, will be `52.5^@` .

Let the slant height be `l` and the vertical height be h.

(a) From the Law of Sines we have `(sin52.5^@)/84.02=(sin32^@)/l`

Then `l=(84.02sin32^@)/(sin52.5^@)~~56.12`

So the slant height is approximately 56.12m

(b) The vertical height can be found by dropping a perpendicular from the vertex of the angle with measure `52.5^@` to the extended side opposite. This forms a right triangle with hypotenuse approximately 56.12m and the angle opposite the vertical leg `84.5^@` .

Then the vertical leg has a length found by:

`sin85.5^@=h/56.12`

So `h=56.12sin84.5^@~~55.86m`

Then the vertical height is approximately 55.86m.

The bottom of the triangle is heading north from right to left; the red perpendicular is dropped from the south side. The angle to the far left is 32 degrees; the angle at the top of the triangle is 52.5 degrees.