Leaning Buildings
The Capital Gate in Abu Dhabi has been certified as the world's "farthest manmade leaning building" because of its 18 degree lean. It's almost shocking to see in person, since it doesn't look like it should be able to stay standing with such a big lean. In fact, a lot of careful planning went into the designing and construction of this 160 m tall building. The foundation contains 490 pilings that extend 20-30 m below ground, providing stability against wind, gravity, and seismic activity. It also has a diagrid (diagonally intersecting metal, which you can see in the photo) designed to keep the building stable.
I wish that we had time to see the view from underneath the leaning part of the building, but we were on our way out of the city, and it would have been difficult to arrange.
We can use the building's height and angle of tilt to calculate how far the roof hangs over the base. We know that the height is 160 m and the angle is 18 degrees. If the distance the roof hangs over the base is given by x, then we have tan(18) = x/160. Solving this gives us x ≈ 28.5 meters, which is quite a bit of distance!
Sample Problems
1. The Leaning Tower of Pisa is the world's most famous leaning tower. It has a height of 55.86 m and leans at an angle of 3.97 degrees. Calculate how far the roof hangs out over the base of the tower.
2. The Gate of Europe towers are a pair of leaning towers in Madrid. Each has a height of 114 m and leans at a 15 degree angle. Calculate how far the roofs hangs out over the bases of the towers.
3. The Veer Towers are a pair of leaning towers in Las Vegas. Each has a height of 150 m, and the roofs lean 13.1 feet over the base of the building. Calculate the angle of lean for the towers.
4. The Dockland Office Building is a leaning tower in Hamburg. It has a height of only about 21 m, but its roof hangs out 47 m over the river Elbe! Calculate the angle of lean of the building.