Arch Section sized to fit within a Rectangular Boundary

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

Circular Arch Section sized to Rectangle ... Definition of Critical Dimensions

Circular Arch Section sized to Rectangle ... Definition of Critical Dimensions

GIVEN DIMENSIONS ...
Outside Radius = 70.25
Arch Depth = 9
Stock Depth = 13
The arch is to be crafted in three equal sections
Find the required Stock Length
Circular Arch Section sized to Rectangle ... Geometry and Algebra (NOT to SCALE)

Circular Arch Section sized to Rectangle ... Geometry and Algebra (NOT to SCALE)

Circular Arch Section sized to Rectangle ... for Three Equal Section Arch

Circular Arch Sections sized to Rectangles ... Three Equal Section Assembly

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
Circular Arch Sections sized to Rectangles ... Five Equal Section Assembly

Circular Arch Sections sized to Rectangles ... Five Equal Section Assembly

Circular Arch Section sized to Rectangle ... for Five Equal Section Arch

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
Circular Arch Section sized to Rectangle ... Solution of Miter Angle and Minimum Outside Radius

Circular Arch Section sized to Rectangle ... Solution of Miter Angle and Minimum Outside Radius

Circular Arch Section sized to Rectangle Formulas

Newton's Method ... Calculator Formula for Miter Angle

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