Gothic Arch

Here's the design and math for our gothic arch greenhouse. We are doing 8 feet by 12 feet with an arch radius of 8 feet * 4/5.

Math!

• let w be the base width of the arch.
  • 8 feet
• let r be the radius of the arch.
  • r = 8 * 4/5
  • r = 6.4 feet
• let a be the distance from the center of the arch to the pivot point
  • a = (w/2) - (w/5)
  • a = (5w/10) - (2w/10)
  • a = 3w/10
  • a = 3(8)/10
  • a = 2.4 feet
• the height (h) of the arch from the ground is sqrt(r^2 - a^2)
  • 5.933 feet
  • this is ignoring the knee height, a vertical wall placed below the arch
• the angle rise from the pivot point to the top of the greenhouse is found using the law of cosines
  • cos^-1((a^2 + r^2 - h^2)/2ar)
  • cos^-1(0.37498)
  • A = 67.977 degrees or 1.1864 radians
• the arch length is A(r), which is 1.186 * 6.4 = 7.593 feet
  • this is the length of wood for the ribs of the greenhouse
• the chord length, which is the line from the top of the greenhouse to a bottom corner, is found using the law of cosines
  • chord length = sqrt(6.2^2 + 6.2^2 - 2(6.2^2)cos(67.877))
  • chord length = 6.68119 feet
  • this is just useful for building the jig and double checking measurments