SAG Calculator

The calculation of sag relies on the fundamental geometric properties of a circle. When you know the radius (R) and the chord diameter or span (D) of an arc, the depth of the sag (S) can be derived using the Pythagorean theorem. The standard equation is expressed as:

sag = radius - sqrt(radius^2 - (diameter / 2)^2)

In this formula, half of the diameter represents the horizontal distance from the center of the span to either endpoint. Squaring this value and subtracting it from the squared radius gives the square of the vertical leg of a right triangle formed by the radius, half-span, and center offset. Taking the square root of this result and subtracting it from the total radius yields the exact sag depth.

Let us walk through a practical scenario to see how the formula operates in the real world. Imagine you are tasked with designing a decorative architectural arch with a span diameter of 10 meters and a radius of curvature measuring 5 meters.

Step 1: Identify your inputs. Diameter (D) = 10m, Radius (R) = 5m.

Step 2: Divide the diameter by 2 to find the half-span. 10 / 2 = 5 meters.

Step 3: Square the radius and the half-span. Radius squared = 5^2 = 25. Half-span squared = 5^2 = 25.

Step 4: Subtract the squared half-span from the squared radius. 25 - 25 = 0.

Step 5: Take the square root of 0, which is 0. Subtract this from the radius: 5 - 0 = 5 meters. In this unique boundary condition where the radius equals half the diameter, the arch forms a perfect semicircle with a sag equal to the radius.

To ensure maximum accuracy when utilizing structural sag calculations in the field, always double-verify your initial field measurements. Even minor discrepancies in span diameter can exponentially skew final curvature requirements. Additionally, account for environmental factors like thermal expansion and material dead load, which can alter the theoretical sag of physical suspension or arch systems over time. Always cross-reference your calculated values with local building codes and engineering safety margins.