How to Find Arch Radius?

Unfortunately, the provided reference is only a snippet of a video about calculating arch radius. Without more information, it's impossible to provide a specific method from that source. However, here's a general overview of how to calculate the radius of an arch:

There are a few common methods to find the arch radius, depending on what information you already have. Here are some approaches:

1. Using the Height and Width (Span) of the Arch

This is a common scenario where you know the total width (span) of the arch and the height (rise) of the arch at its highest point.

• Variables:

  • R = Radius of the arch
  • W = Width (span) of the arch
  • H = Height (rise) of the arch

• Formula:

  R = (H/2) + (W^2 / (8 * H))

  This formula allows you to calculate the radius directly from the width and height of the arch.

• Example:

  Let's say you have an arch that is 6 feet wide (W = 6) and 1 foot high (H = 1). Then,

  R = (1/2) + (6^2 / (8 * 1))
  R = 0.5 + (36 / 8)
  R = 0.5 + 4.5
  R = 5 feet

  Therefore, the radius of the arch is 5 feet.

2. Using a Chord and Sagitta

• Chord: A straight line connecting two points on the arch. The width of the arch in the previous example is a chord.
• Sagitta: The perpendicular distance from the midpoint of the chord to the arch. This is the height in the previous example.

This method is essentially the same as method 1, just described with different terminology.

3. Geometrically (Practical Method)

If you have an actual arch (or a drawing of one), you can find the radius geometrically:

1. Draw the arch: Have the arch you want to measure.
2. Draw a chord: Select any two points on the arch and connect them with a straight line (the chord).
3. Find the midpoint of the chord: Measure the length of the chord and find the midpoint.
4. Draw a perpendicular line: Draw a line perpendicular to the chord, passing through its midpoint, and extending to intersect the arch (this is the sagitta).
5. Extend the perpendicular line: Extend the perpendicular line (sagitta) beyond the arch.
6. Use a compass: Place the point of a compass at the top of the arch (where the sagitta intersects the arch). Adjust the compass's radius until the pencil touches one end of your chord. Swing the compass to see where it intersects the extended sagitta line. The center of your radius will be at the compass point on this line.
7. Measure: Measure from the starting point on the extended sagitta line (where it meets the base of the arch) to the intersecting radius point. This will be the radius.

Considerations

• Accuracy: The accuracy of the calculated radius depends on the accuracy of your measurements (width and height).
• Type of Arch: These methods generally apply to circular arches. For other types of arches (e.g., elliptical, parabolic), the calculations are more complex.