Circular Arch Section
within a
Rectangular Boundary
Cubic Equation Solution
Rectangular Stock Length =
Rectangular Stock Depth =
Circular Arch Depth =

General Cubic Equation ...
Ax3 + Bx2 + Cx + D = 0

A =
B =
C =
D =

Copy and paste to WZ Function Grapher
created by Walter Zorn

Let  B/A = p, C/A = q and  D/A = r ...
x3 + px2 + qx + r = 0

p =
q =
r =

Let  x = x1 – p/3 ...
x13 + ax1 + b = 0

a =
b =
– b/2 =
a³/27 + b²/4 =
√|a³/27 + b²/4| =
√(a³/27 + b²/4) =
– b/2 + √(a³/27 + b²/4) =
– b/2 – √(a³/27 + b²/4) =
x1 =

x = x1 – p/3 ...

Outside Radius =
Radians ... Miter Angle =
Degrees ... Miter Angle =
Outside Radius ... Sagitta =
Inside Radius ... Sagitta =
Bottom Stock Length =
Circular Arch Section sized to Rectangle ... Definition of Critical Dimensions
In the following formulas, "Radius" means the Outside Radius
Trigonometric Forms of Equations
Squaring the terms, and grouping terms according to the powers of the Radius
Terms squared and collected in terms of powers of the Radius
Coefficients of Powers of the Radius
Solution of the Radius
Cubic Equation Solver

General Cubic Equation ...
Ax3 + Bx2 + Cx + D = 0

A =
B =
C =
D =

Let  B/A = p, C/A = q and  D/A = r ...
x3 + px2 + qx + r = 0

p =
q =
r =

Let  x = x1 – p/3 ...
x13 + ax1 + b = 0

a =
b =
– b/2 =
a³/27 + b²/4 =
√|a³/27 + b²/4| =
√(a³/27 + b²/4) =
– b/2 + √(a³/27 + b²/4) =
– b/2 – √(a³/27 + b²/4) =

If  a³/27 + b²/4 < 0 ...

Modulus r =
Argument θ =
θ ... + i
2π – θ ... + i
2xreal – p/3 =
θ + 2π/3 ... + i
2π – θ – 2π/3 ... + i
2xreal – p/3 =
θ + 4π/3 ... + i
2π – θ – 4π/3 ... + i
2xreal – p/3 =

If  a³/27 + b²/4 ³ 0 ...

x1 =
x = x1 – p/3 =

Joe Bartok