How to calculate an arch doorway?

To calculate an arch doorway's approximate arc length, you'll need its opening's width and height, applying a specific method involving the Pythagorean theorem. This calculation helps you determine an estimated length of the curved part of your arch.

Calculating Arch Doorway Dimensions

Follow these steps to calculate the approximate arc length for your arch doorway based on its opening dimensions:

Step 1: Measure Opening Dimensions

Begin by accurately measuring the dimensions of the doorway opening where the arch will be installed.

• Width: Measure the horizontal distance across the opening.
• Height: Measure the vertical distance from the floor (or the base of the opening) to the top of the opening.

Step 2: Determine Arch Radius

Once you have the height measurement, you will use it to determine the radius of the arch.

• Formula: Divide the measured height of the opening by 2.
• Purpose: This calculated value will serve as one of the sides of the right triangle used in the next step.

Step 3: Form a Right Triangle

To apply the Pythagorean theorem, you need to define the two legs of a right triangle relevant to your arch.

• Side 1: This side will be equal to half of the total width of your doorway opening.
• Side 2: This side will be the radius you determined in Step 2 (which is half of the opening's height).

Step 4: Calculate Approximate Arc Length

With the two sides of your right triangle defined, you can now use the Pythagorean theorem (a² + b² = c²) to find the hypotenuse, which, according to this method, will give you an approximate arc length for your door.

• Variables:
  • a = Half of the doorway's width
  • b = The calculated radius (half of the doorway's height)
  • c = The hypotenuse, representing the approximate arc length
• Formula: Approximate Arc Length (c) = √ ( (Width/2)² + (Height/2)² )

Practical Example: Calculating Your Arch

Let's walk through an example to illustrate the process:

Suppose you have a doorway opening with the following measurements:

Now, let's apply the steps:

1. Measure Opening Dimensions:

   • Width = 36 inches
   • Height = 80 inches

2. Determine Arch Radius:

   • Radius = Height / 2
   • Radius = 80 inches / 2 = 40 inches

3. Form a Right Triangle:

   • Side 1 (a) = Width / 2 = 36 inches / 2 = 18 inches
   • Side 2 (b) = Radius = 40 inches

4. Calculate Approximate Arc Length:

   • Approximate Arc Length (c) = √ ( a² + b² )
   • c = √ ( (18)² + (40)² )
   • c = √ ( 324 + 1600 )
   • c = √ ( 1924 )
   • c ≈ 43.86 inches

Therefore, for a doorway opening of 36 inches wide and 80 inches high, the approximate arc length for your door, according to this calculation method, would be approximately 43.86 inches.