Arch Calculator
Calculate elliptical arch focus points and measurements for construction and drawing
Calculate Elliptical Arch Parameters
The horizontal distance between the two vertical supports
The vertical distance from the base to the highest point
Example Calculation
Doorway Arch Example
Door opening: 880mm wide (typical door width)
Desired rise: 300mm (attractive proportion)
Application: Interior decorative archway
Calculation
a = 880mm ÷ 2 = 440mm (semi-major axis)
b = 300mm (semi-minor axis)
Focal Distance = √(440² - 300²) = √(193,600 - 90,000) = √103,600
Focal Distance = 321.9mm
Focus points: ±321.9mm from center
Types of Arches
Elliptical
Oval shape, adjustable rise
Best for custom proportions
Semi-circular
Half circle, Roman style
Rise = half the span
Segmental
Segment of circle arc
Low-rise design
Tools Needed
Measuring tape or ruler
90° square or T-square
Pencil or marking tool
String or twine
Three nails or pins
Understanding Elliptical Arches
What is an Elliptical Arch?
An elliptical arch is shaped as half of a horizontal ellipse. Unlike circular arches, elliptical arches allow you to customize both the width (span) and height (rise) independently, making them perfect for architectural applications where proportions matter.
Why Use Elliptical Arches?
- •Adjustable proportions for any space
- •Lower horizontal loads than segmental arches
- •Elegant visual appeal
- •Suitable for both decorative and structural use
Mathematical Foundation
Focal Distance = √(a² - b²)
- a: Half the arch length (semi-major axis)
- b: Arch rise height (semi-minor axis)
- F1, F2: Focus points ±focal distance from center
- Property: Sum of distances from any point to both foci equals 2a
Construction Principle: String length = arch length. Fixed at focus points, the string traces a perfect elliptical arc when kept taut.
String Method Drawing Guide
Step-by-Step Instructions
Draw Base Line
Mark a straight line equal to your arch length
Mark Center
Find and mark the midpoint of your base line
Locate Focus Points
Measure focal distance left and right of center
Draw Height Line
Draw perpendicular line for arch rise from center
Set Up String
Place nails at focus points and top, attach string
Draw Arch
Remove top nail, keep string taut, trace the curve
Pro Tips
String Length
Use string length equal to arch base length for perfect ellipse
Accuracy
Keep string taut and perpendicular to drawing surface
Material Choice
Non-stretch string works best - fishing line or mason's line
Large Arches
For big projects, use screws instead of nails for stability
Important Note
For structural arches, consult a structural engineer. This calculator is for geometric construction only.