Arch Geometry: Span, Rise, and Radius
This calculator uses basic geometry to derive the full arc properties from the two primary measurements of a **circular arch segment**: the **Span ($S$)** and the **Rise ($R$)**. These calculations are essential for masonry, carpentry, and architectural design to ensure structural integrity and correct dimensions.
The Key Formula: Finding the Radius
The Radius ($\mathbf{r}$) is the most critical component, derived from the Pythagorean theorem applied to the triangle formed by the circle's center, the arc's peak, and the springing point.
$$\mathbf{r} = \frac{(\mathbf{S}/2)^2 + \mathbf{R}^2}{2 \times \mathbf{R}}$$
Understanding Arch Terms
- **Span ($S$):** The horizontal distance between the arch supports (**springing points**). Also known as the chord length.
- **Rise ($R$):** The vertical distance from the center of the span up to the highest point of the arch (**crown**).
- **Arc Length ($L$):** The length of the curved line itself, which is needed to calculate the amount of material required for the curve (e.g., molding, stone blocks).
- **Springing Angle ($\Phi$):** The angle the arch forms with the horizontal at the springing points. This is crucial for load distribution and determining the necessary buttressing or support structure.
Note on Arch Types
This calculator applies specifically to **circular arches** (segmental arches). Other types, like pointed Gothic arches (ogival) or elliptical arches, require different, more complex formulas for accurate calculation.