Circular Arch Section
sized to fit within a
Rectangular Boundary

Linear Equation Solutions ... Circular Arch Calculations and Drawings
Graphical Solutions ... Circular Arch Calculations and Drawings

GIVEN DIMENSIONS ...
Arch Depth = 9
Stock Depth = 13
The arch is to be crafted in three equal sections
Find the required Stock Length

GIVEN DIMENSIONS ...
Arch Width = 119.518146
Arch Height = 33.318771
Arch Depth = 9
The arch is to be crafted in five equal sections
Solve the Outside Radius and required Stock Dimensions

 Angles used in calculations are be expressed in radians Outside Radius = (119.518146² + 4 × 33.318771²)/(8 × 33.318771) = 70.25 Arc Length = 2 × 70.25 × arcsin ((.5 × 119.518146)/70.25) = 142.922934 Check: Circular Arch Calculator … enter Arch Width and Arch Height The arc length and angle are divided into five equal parts: Section Arc Length = 142.922934/5 = 28.584589 Miter Angle = arcsin ((.5 × 119.518146)/70.25))/5 = .203449016 Miter Angle = .203449016 × 180/pi = 11.656770° … starts at the 1.448869 sagitta depth* Stock Length = Chord = 2.× 70.25 × sin .203449016 = 28.387801 Sagitta = 70.25 × (1 – cos .203449016) = 1.448869* Check: Sagitta Calculator … enter Section Arc Length and Chord (Stock Length) Height from Circle Center = 70.25 – 10.448869 = 59.801131 Inside Radius of Arch = 70.25 – 9 = 61.25 Height of Inside Radius from Circle Center = 61.25 × cos .203449016 = 59.986752 Height of Inside Radius from Bottom of Timber = 59.986752 – 59.801131 = .185621 Stock Depth = 9 + 1.448869 – .185621 = 10.263248 Dimension of Inside Radius from Arch Center = 61.25 × sin .203449016 = 12.375465 Check: Bow Window Calculator … enter: Width or Span = 119.518146 Projection = 33.318771 – 1.448869 = 31.869902 Number of Bays (the number of arch sections) = 5

GIVEN DIMENSIONS ...
Stock Length = 28.387801
Stock Depth = 10.263248
Arch Depth = 9
Determine the Miter Angle and the Minimum Outside Radius

The original version of this graph was plotted using the
WZ Function Grapher created by Walter Zorn

Newton's Method ... Miter Angle and Best Fit Radius
Cubic Equation Solver ... Best Fit Radius