Arch Calculator
Calculate dimensions, area, and brick requirements for different types of arches
Arch Calculator
Table of Contents
Types of Arches
Arches have been used in architecture for thousands of years, providing both structural support and aesthetic appeal. Our calculator supports the following common arch types:
Semicircular Arch
A semicircular arch forms a perfect half-circle, with its height equal to half its width (span). This is one of the oldest and most common arch types, widely used in Roman architecture.
- Height always equals half the span
- Radius equals half the span
- Distributes weight evenly to the supports
- Relatively simple to construct
- Requires substantial height clearance
Key Formulas:
Radius (r) = Span (w) ÷ 2
Height (h) = Radius (r)
Area = (π × r²) ÷ 2
Perimeter = π × r + w
Segmental Arch
A segmental arch is a portion (segment) of a circle with a height less than half its width. The center point of the circle lies below the spring line (base level) of the arch.
- Height is less than half the span
- Radius is greater than half the span
- Requires less vertical space than semicircular arches
- Common in bridges and modern architecture
- Creates horizontal thrust that must be counteracted
Key Formulas:
Radius (r) = (h² + (w/2)²) ÷ (2h)
Area = (r² × θ) ÷ 2
where θ = angle in radians
θ = 2 × sin⁻¹(w ÷ (2r))
Gothic Arch
A gothic arch consists of two circular segments that meet at a point at the top. This arch type was extensively used in medieval architecture, particularly in cathedrals.
- Height is typically equal to or greater than the span
- Creates a pointed apex
- Excellent for distributing vertical loads
- Allows for greater height and larger windows
- More complex to construct than semicircular arches
Key Formulas:
Approximate area ≈ (2/3) × w × h
Approximate perimeter ≈ 2 × √((w/2)² + h²)
Note: Gothic arches vary in exact geometry; these are simplified approximations.
How to Calculate Arch Dimensions
Calculating arch dimensions involves understanding the geometric relationships between span, height, radius, and other parameters. Our calculator offers two methods for defining an arch:
Width & Height Method
This method uses the horizontal span (width) and the vertical rise (height) to define the arch. It's the most intuitive approach when you know the opening dimensions.
For Semicircular Arches:
- The height must equal half the width
- Radius = Width ÷ 2
For Segmental Arches:
- The height must be less than half the width
- Radius = (Height² + (Width/2)²) ÷ (2 × Height)
- Angle (radians) = 2 × sin⁻¹(Width ÷ (2 × Radius))
For Gothic Arches:
- The height is typically equal to or greater than the width
- Exact calculations vary based on the specific gothic style
Width & Radius Method
This method uses the horizontal span (width) and the radius of the circular segments to define the arch. It's useful when designing with specific structural requirements in mind.
For Semicircular Arches:
- The width must equal 2 times the radius
- Height = Radius
For Segmental Arches:
- The radius must be greater than half the width
- Height = Radius - √(Radius² - (Width/2)²)
- Angle (radians) = 2 × sin⁻¹(Width ÷ (2 × Radius))
For Gothic Arches:
- Typically uses two radius centers positioned along the spring line
- The radius is often equal to the width for equilateral gothic arches
Important Measurements:
- Span/Width: The horizontal distance between the two supports of the arch
- Rise/Height: The vertical distance from the spring line to the highest point of the arch
- Radius: The radius of the circle(s) that form the arch curve
- Thickness: The depth of the arch perpendicular to its face
- Spring Line: The horizontal line at the level where the arch begins to curve
- Intrados: The inner curve of the arch
- Extrados: The outer curve of the arch
Calculating Brick Requirements
Determining the number of bricks needed for an arch is essential for material planning and cost estimation. Our calculator provides an estimate based on standard brick dimensions and mortar joints.
Brick Layout Considerations
When building a brick arch, several factors affect the brick count and layout:
- Voussoirs: Wedge-shaped bricks specifically designed for arches
- Rowlock Course: Bricks laid on edge with the narrow face showing
- Soldier Course: Bricks laid vertically with the narrow face showing
- Header Course: Bricks laid with the end face showing
- Stretcher Course: Bricks laid with the long face showing
The most common brick arch construction methods include:
- Rowlock Arch: Uses standard bricks laid on edge, with mortar joints tapered to create the arch shape
- Voussoir Arch: Uses specially shaped wedge bricks designed to fit the arch curve
- Corbeled Arch: Uses overlapping bricks that gradually project further with each course
Standard Brick Dimensions
| Brick Type | Length (mm) | Width (mm) | Height (mm) |
|---|---|---|---|
| Standard (UK) | 215 | 102.5 | 65 |
| Standard (US) | 203 | 102 | 57 |
| Modular (US) | 194 | 92 | 57 |
| Engineering (US) | 203 | 102 | 76 |
Note: Actual brick dimensions may vary by manufacturer and region. Always check the specific dimensions of the bricks you plan to use.
Mortar Joint Considerations
Mortar joints in arch construction typically range from 8-12mm (3/8"-1/2"). For arches, joints are often tapered:
- Wider at the extrados (outer curve)
- Narrower at the intrados (inner curve)
- This tapering helps create the curved shape when using standard rectangular bricks
When calculating brick quantities, include the mortar joint thickness in your calculations. Our calculator adds the joint thickness to the brick dimensions to determine the effective area covered by each brick.
Brick Count Formula:
Brick Count = Arch Surface Area ÷ Effective Brick Area
Where:
- Arch Surface Area = Arch Perimeter × Arch Thickness
- Effective Brick Area = (Brick Length + Joint Thickness) × (Brick Height + Joint Thickness)
Note: Add 10-15% to the calculated brick count for waste, cutting, and breakage.
Arch Construction Tips
Building a successful arch requires careful planning and execution. Here are some practical tips for arch construction:
Formwork and Templates
Proper support during construction is essential for arch stability:
- Create a sturdy wooden formwork (centering) that matches the exact curve of your arch
- Ensure the formwork can support the full weight of the arch until the mortar cures
- For precise curves, create templates based on the calculated radius
- Don't remove formwork until the mortar has fully cured (typically 7-14 days)
- When removing formwork, start from the center and work outward
Keystone and Voussoirs
The arrangement of arch stones or bricks is critical to structural integrity:
- The keystone (center stone) should be placed last to lock the arch in place
- All voussoirs (arch stones/bricks) should be oriented toward the center point(s) of the arch
- For brick arches without custom voussoirs, cut standard bricks to create tapered joints
- Ensure even pressure distribution by maintaining consistent joint thickness at the intrados
Structural Considerations
Arches create lateral thrust that must be properly managed:
- Ensure adequate support at the spring points to resist horizontal thrust
- For larger arches, consider buttresses or tie rods to counteract lateral forces
- The thickness of the arch should increase proportionally with the span
- For load-bearing arches, consult a structural engineer for proper sizing
- Allow for settlement by slightly overbuilding the arch height (camber)
Mortar Selection
The right mortar mix is crucial for arch durability:
- Use Type N mortar for most decorative and low-load applications
- Use Type S mortar for structural and load-bearing arches
- Consider using a slightly stiffer mortar mix for arches to reduce slumping
- For historical restorations, match the original mortar composition
- Ensure consistent mortar mixing throughout the project
Professional Assistance:
While small decorative arches can be DIY projects, consider professional help for:
- Load-bearing arches
- Spans greater than 1.2 meters (4 feet)
- Complex arch designs (gothic, multicentered, etc.)
- Structural modifications to existing buildings
Always check local building codes and consider consulting a structural engineer for any arch that will support significant loads.
Frequently Asked Questions
What is the difference between a semicircular and segmental arch?
A semicircular arch forms a perfect half-circle with the height equal to half the width (span). It has a consistent radius throughout, with the center point at the spring line. A segmental arch, on the other hand, is a portion (segment) of a circle with a height less than half its width. The center point of a segmental arch is below the spring line, and it forms a flatter curve than a semicircular arch. Segmental arches are often used when height constraints exist or for aesthetic preferences.
How do I determine the right type of arch for my project?
Choosing the right arch type depends on several factors: 1) Structural requirements - semicircular arches distribute weight most efficiently, while gothic arches excel at height; 2) Available height - segmental arches require less vertical space; 3) Aesthetic preferences - gothic arches create a more dramatic, vertical appearance while semicircular arches offer a classic look; 4) Historical context - if matching existing architecture, choose the style that complements the surrounding structures; 5) Construction complexity - semicircular arches are generally simpler to build than gothic arches. Consult with a structural engineer for load-bearing applications.
Why is the thickness of an arch important?
The thickness of an arch is crucial for its structural integrity and stability. An arch works by converting vertical loads into compressive forces that travel through the arch to its supports. Adequate thickness ensures the arch can handle these compressive forces without failing. The proper thickness depends on the span of the arch, the materials used, and the load it needs to support. As a general rule, the minimum thickness at the crown (top) of a masonry arch should be at least 1/18 to 1/24 of the span for semicircular arches, and slightly more for segmental arches. For structural applications, always consult with a structural engineer to determine the appropriate thickness.
How do I calculate the number of bricks needed for an arch?
To calculate the number of bricks for an arch: 1) Calculate the surface area of the arch (perimeter × thickness); 2) Determine the effective area covered by one brick including mortar joints (brick length + joint thickness) × (brick height + joint thickness); 3) Divide the arch surface area by the effective brick area to get the approximate brick count; 4) Add 10-15% for waste and cutting. For wedge-shaped arches, specialized voussoir bricks or cut bricks may be needed, which affects the count. Our calculator provides an estimate, but actual requirements may vary based on the specific construction method and bond pattern used.