Construction Math Formulas, Diagrams and Calculators

Hip and Valley Roof Framing Angles
... formulas, diagrams and example solutions

Hip and Valley Roof Framing and Joinery Math
Framing Angle Calculator and Diagrams
Framing and Joinery Angle Formula Catalogue
Hip and Valley Roof: Basic Geometric Models
3D Hip Roof Models and Trigonometry (pdf)
Fundamental Hip Roof Formulas
Overview of Tetrahedra and Triangles defining a Hip or Valley Roof
Roof Framing Angles: Formulas and Sketches
Irregular Valley Roof Sketches
Valley Length between Working Points
Irregular Plan Angle Formula
Irregular Plan Angle and Hip Roof Calculations
Construction of the Backing Angle
Backing Angle: Regular Octogonal Gazebo
Studies of some Backing Angle Formulas and Geometry
Backed Valley Cross Sections
Backing Angles on Log Valley Rafters (pdf)
Tetrahedron modeling Backing Angle extracted from Hip Rafter
Proportional Hip Shift and Hip Drop (pdf)
Example Hip Shift-Drop Calculations
Example solutions ... Irregular Bay Roof with Overhangs
Dormer with Sloped Ridge
Sloped Roof with "Star" Footprint
Cricket Valley Calculations
Clipped Corner Hip Roof Calculations
Study of a Prow Peak or Rake Wall
Prow Peak Sketches and Calculations
Prow Peak compared to Valley Rafter
Valley Rafter intercepts Beam
Valley Soffit Overhang Adjustment
HAP Calculations and Sketches
Square Fascia meets Hip-Valley Foot
Square Fascia meets Main Roof (any Ridge intercept Angle)
Square Fascia meets Main Roof (90° Ridge Intercept)
Gazebo with Square Tail Fascia Hip Rafters
Layover or Sleeper Valley Length Adjustments
Angled Rafter or Timber intersects Sloped Wall
Jack Rafters intersect "Layover Valley" at Irregular Plan Angle
Compound Angle for Louvered Vent
Compound Angle for Hopper or recessed Housing
Hip Roof Ridge Cap
Study of a Compound Joint

Hip and Valley Roof Developmental Geometry

Slideshow ... Development of Tetrahedron composed of Right Triangles
Trigonometric Relationships between Tetrahedron Angles
Developed Model of Hip Roof
Developments of Hip and Valley Roof Angles based on the Roof Lines
Arrangements of Fundamental Hip Roof Angles
Developed Model of Compound Angle
Developed Tetrahedron: Jack Rafter Compound Angle
Hip-Valley Side Cut Angle Developments
Developments of Hip Side Cut Angle and Backing Angle Tetrahedra
Developed Tetrahedron: Purlin Compound Angle
Development of Purlin Related Angles (pdf)
Developments of "Layover" type Rafters and Purlins
Irregular Hip ... Development of Square Hung Fascia Compound Angle
Regular Hip ... Development of Square Hung Fascia or Purlin Compound Angle
Development of Polyhedron extracted from the Hip Rafter
Compound Rake or Prow Roof Crown Molding or Fascia
Interlocking Tetrahedra ... 3D Model of Backing Angle
Interlocking Tetrahedra ... 3D Model of Stud Compound Angle for Prow Peak
Interlocking Tetrahedra ... 3D Model of Chamfer Cut (Square Cut) Angles

Roof Framing Angle Geometry: Animations and Slideshows

Development of Saw Blade Angle (slideshow)
Tetrahedral Model of Compound Angle (slideshow)
Compound Angle and Hip Roof Models (slideshow)
Development of Backing Angles (slideshow)
Tetrahedral Model of Backing Angle (slideshow)
Revolution of the Plane of the Backing Angle (slideshow)
Development of Proportioned Plumb Backing Angles (slideshow)
Sheathing Angle Ratio and Formula (slideshow)
Development of Angle on Upper Shoulder of Hip-Valley (slideshow)
Blade Bevel Development - Upper Shoulder of Hip-Valley (slideshow)
Development of Purlin Related Angles (slideshow)
Housing Chamfer or Shelf Cut (slideshow)
Hip Rafter : Development of Square Tail Fascia Angles (slideshow)

Curves, Arches and Intersections of Curved Surfaces

Cross Vault: Intersecting Elliptical Prisms
Intersecting Barrels with Dissimilar Profiles
Development of Arch intercepts Cove
Development of Arch intersects Main Roof
Intersection of Cone and Slope of Roof
Development of Arch or Barrel intersecting a Conical Roof
Hip Roof with flared Rafters and Eaves
Graphing and Lofting Ellipsoids and Ovoids
Development of Cylindrical Flashing intersection with Slope of Roof
Circular Arc tangent to intersection of Rafters
Circular Arch sized to Rectangular Boundary
Miscellaneous Ellipse Formulas and Arch Layout Notes
Pyramid or Cone with a sloped Base
Bell Curve ... Eyebrow Dormer Profile
Grahpical Solutions of Bowed Bay Window Central Angle
Studies of Tangent Handrailing Mathematics

Miscellaneous Math Notes

Miscellaneous Ellipse Formulas
Eccentric Angle of Ellipse
Slope of Ellipse at any Point
Formula for the Slope of an Ellipse
Simpson's Rule: Ellipse Arc Length between Two Points
Simpson's Rule ... estimation of Ellipse Arc Length
Convergent Series ... Tangent of an Angle
Solution of Arcsin Series and Evalution of π
Square Root: Newton's Method and Binomial Theorem
Newton's Method: Solving Circle Radii given Arc Lengths
Newton's Method: Radius of Two Chord Circle
Newton's Method: Circular Arch sized to Rectangular Boundary
Newton's Method: Solution of Bow Window Central Angle
Geometric interpretation of Trig Functions
Helix Radius of Curvature Formulas
Studies of Geometric Solids

Javascript Calculators

Framing Angle Calculator and Diagrams
Hip and Valley Dimensioning Calculator
Square Tail Fascia intercepts Hip Rafter
Purlin/Square Tail Fascia intercept Main Roof
Jack Rafters intersect Layover Valley Calculator
Sloped Frustum of a Pyramid or Cone Calculator
Cross Vault Purlin Calculator
Bell Curve Calculators ... plot Eyebrow Dormer Profile
Dovetail Calculator
Linear Equation Calculators
Circle Calculators
Circle Center given Three Points on the Circle
Sagitta given Arc Length and Chord
Rectangular Hyperbola Calculator
Helix Analysis
Torus and Elliptical Surfaces of Revolution
Ellipsoid and Ovoid Calculator
Parabolic Surface of Revolution
Hyperbolic Surface of Revolution
Circular Arch Calculator
Elliptical Arch Calculator (known: Major or Minor Axis)
Elliptical Arch Calculator (known: two Arch Dimensions)
Ellipse Axes Calculator (known: two Points on Ellipse)
Ellipse intersects Line Calculator
Plot an Oval parallel to an Ellipse
Hyperbolic Arch Calculator
Square Fascia meets Main Roof (90° Ridge Intercept)
Rafter at Fixed Ridge Height Analysis
Hip and Valley Sections
Roof with Elliptical Footprint Calculator
Regular Polygon Dimensions Calculator
Bow Window Radius and Central Angle Calculator
Bow Window Side Length Tester (enter Trial Angle)
Third Degree or Cubic Equation Solver: Circular Arch sized to Rectangular Boundary

Excel Worksheets

Force Solver : Cramer's Rule (xls)
Circular Arc tangent to Rafter Foot (xls)
Square Tail Fascia intercepts Hip Rafter (xls)
Purlin intercepts Main Roof Calculator (xls)
Purlin Calculator for Sloped Ridge Lines (xls)
Cross Vault Calculator (xls)
Arch intercepts Main Roof (xls)
Unadjusted Hip-Valley Imperial Dimensions (xls)

Links to Mathematics Websites

Send your comments or questions ... Joe Bartok