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What is the Equation of a Circular Arch?

Published in Circular Arch Geometry 5 mins read

The equation of a circular arch is derived directly from the fundamental definition of a circle. A circular arch is essentially a specific segment or portion of a full circle.

The exact answer, based on the provided reference, is:

A circular arch is defined by the same foundational equation as a complete circle:

(x−h)² + (y−k)² = r²

Where:

  • (h,k) represents the coordinates of the center of the full circle from which the arch segment is taken.
  • r is the radius of that circle.

To specifically describe the curve of the arch, this equation is typically rearranged to express y in terms of x (or x in terms of y), along with defined constraints on the domain (x-values) or range (y-values) that limit it to the arch segment.

Understanding the Circular Arch Equation

While the base equation is that of a circle, the term "arch" implies a specific segment. This segment usually has a defined horizontal "span" and a vertical "rise."

When solving the circle equation for y, you get:

(y−k)² = r² − (x−h)²
y − k = ±√(r² − (x−h)²)
y = k ±√(r² − (x−h)²)

The choice of + or - sign determines whether the arch represents the upper or lower segment of the circle relative to its center (h,k). For a typical upright arch (like a bridge or a doorway), the y values are usually positive.

Key Parameters for Defining a Circular Arch

To fully define a circular arch, several key parameters are interconnected:

  • Center of the Circle (h, k): The origin point of the full circle.
  • Radius (r): The distance from the center to any point on the arch.
  • Span (W): The horizontal distance between the two base points of the arch.
  • Rise (H): The vertical distance from the base line to the apex (highest point) of the arch.

These parameters are interdependent. For example, if an arch is centered on the y-axis (h=0) and its base rests on the x-axis (y=0), the relationship between its radius, half-span, and the y-coordinate of its center can often be found using the Pythagorean theorem.

Common Scenarios and Practical Insights

Circular arches are frequently designed to be symmetrical, often centered on the y-axis (h=0). Here are common ways their equations are formulated:

Scenario 1: Arch Opening Upwards (Center Below Base)

This configuration is typical when the full circle's center (0, k) is below the base of the arch. The arch represents the upper portion of the circle.

  • General Form: x² + (y − k)² = r²
  • Solving for y (Arch Segment): y = k + √(r² − x²)
    • Constraints: y ≥ 0 (if the base is at y=0), and typically −W/2 ≤ x ≤ W/2, where W is the arch's span.
  • Practical Insight: In this setup, if the arch's base is at y=0, then r² = (W/2)² + k². The rise H of the arch above the base would be r + k (since k would be a negative value).

Scenario 2: Arch Opening Downwards (Center Above Base)

This is a very common representation for architectural arches where the full circle's center (0, k) is above the base of the arch. The arch represents the lower portion of the circle.

  • General Form: x² + (y − k)² = r²
  • Solving for y (Arch Segment): y = k − √(r² − x²)
    • Constraints: y ≥ 0 (if the base is at y=0), and typically −W/2 ≤ x ≤ W/2.
  • Practical Insight: If the arch's base is at y=0, then the y-coordinate of the center k equals the radius r of the circle. In this case, the equation simplifies to x² + (y − r)² = r². The total rise H of the arch is r, and the total span W is 2r.

Example: Calculating the Equation for a Specific Arch

Let's determine the equation for a circular arch with a span of 8 meters and a rise of 3 meters, assuming its base is on the x-axis and it is symmetrical about the y-axis.

  1. Identify Knowns:

    • Span (W) = 8 m, so half-span (x_max) = 4 m. Base points are (-4, 0) and (4, 0).
    • Rise (H) = 3 m. Apex is (0, 3).
    • Center (h, k) = (0, k).
    • Equation form: x² + (y - k)² = r²
  2. Use Points to Find k and r:

    • Point (4, 0) lies on the circle:
      4² + (0 - k)² = r²
      16 + k² = r² (Equation 1)
    • Point (0, 3) (the apex) lies on the circle:
      0² + (3 - k)² = r²
      (3 - k)² = r² (Equation 2)
  3. Solve for k:
    Equate (1) and (2):
    16 + k² = (3 - k)²
    16 + k² = 9 - 6k + k²
    16 = 9 - 6k
    7 = -6k
    k = -7/6 ≈ -1.167 meters

  4. Solve for (and r):
    Using k in Equation 2:
    r² = (3 - (-7/6))² = (3 + 7/6)² = (18/6 + 7/6)² = (25/6)²
    r² = 625/36 ≈ 17.361
    r = 25/6 ≈ 4.167 meters

  5. Formulate the Arch Equation:
    The center of the circle is (0, -7/6). The radius is 25/6.
    The full circle equation is:
    x² + (y - (-7/6))² = (25/6)²
    x² + (y + 7/6)² = 625/36

    Since the arch is the upper segment of this circle (y-values are positive while the center's y-coordinate is negative), we solve for y and take the positive root:
    (y + 7/6)² = 625/36 − x²
    y + 7/6 = √(625/36 − x²)
    y = √(625/36 − x²) − 7/6

    Constraints: This specific arch is valid for −4 ≤ x ≤ 4.

This example illustrates how the general circle equation is applied and constrained to precisely define a circular arch based on its desired dimensions.