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How to Calculate Arch

If you are remodeling your home, it is probably because you trying to improve upon what you already have. Therefore, it is very important that all your calculations be correct, when it comes to installing doors, windows and other fixtures. An arched doorway or window provides an aesthetically pleasing alternative to the traditional rectangular style, but there are a few more calculations involved in order to make sure your arch has the proper dimensions. If done properly, you can end up with something very impressive.

Arches are a visually appealing touch in architecture.
  1. Measure the height and width of the arch. The height should be measured up to the point where the arch starts curving. The difference between this number and the height at the center of the arch is called the rise. For example, if the sides are 72 inches tall, and the center of the arch is 84 inches tall, the rise would be 12 inches.

  2. Use the formula Rise^2 + (Span/2)^2 / 2 x Rise to find the radius of the arch. Plug in your numbers, and use the order of operations PEMDAS (parentheses, exponents, multiplications, addition, subtraction) to solve the equation. In the example above, we had a rise of 12 inches. Let's say that the width (or span) is 60 inches. Therefore, the equation becomes 12^2 + (60/2)^2 / 2x12.

  3. Calculate the parentheses first. 60/2 = 30, so the equation becomes 12^2 + (30)^2 / 2 x 12.

  4. Do the exponents next. 12^2 = 144, and 30^2 = 900. The equation should now look like this: 144 + 900 / 12 x 2.

  5. Multiplication is next. 12 x 2 = 24, so we have 144 + 900 / 24.

  6. Add together the numbers on the top of the fraction. 144 + 900 = 1044. 1044/24 can be reduced to 43.5. The radius of the arch is 43.5 inches.

About the Author

Katie Jacobson has been writing since 2008. Her writing is also featured in valeveil’s "CASEBOK/Ekphrasis" project, as well as on 2ktwelve.com and in "NextWords." She holds a B.A. in English from Occidental College and a writing M.F.A. from CalArts.