Arch Calculator

Find the radius and arc length of a circular arch from its width and height.

How the Math Works

The Arch Calculator uses the formula radius = (span² + 4·rise²) ÷ (8·rise) to determine the radius of a circular arch. This equation derives from the geometric relationship between the arch's span (chord length) and rise (sagitta). When you draw a radius to the arch's highest point, it forms a right triangle with half the span and the difference between the radius and rise. By applying the Pythagorean theorem and rearranging terms, the formula simplifies to the given expression, allowing precise calculation of the circle's radius from two key measurements.

Practical Applications

This calculation is essential in architecture and construction when designing or building arches for bridges, doorways, windows, or decorative elements. Knowing the radius enables accurate placement of centering during construction or precise cutting of materials like stone, brick, or metal. Additionally, the arc length can be derived once the radius is known, helping estimate material quantities needed for curved structures while ensuring structural stability and aesthetic proportions in engineering projects.

Day-to-Day Use

For DIY enthusiasts, this calculator aids in creating home projects like garden trellises, pergolas, or curved furniture where precise measurements matter. It also helps in everyday scenarios such as planning curved pathways, estimating rope or fencing lengths for arched shapes, or even in art projects requiring symmetrical curves. By simplifying complex geometric calculations, it empowers individuals to confidently tackle tasks involving curved designs without specialized tools or expertise.

Worked example

8 ft span, 3 ft rise → radius 4.17 ft, arc ~9.8 ft.

FAQ