How do you calculate the focal points of an arch?

The focal distance is calculated using the formula f = sqrt(a² - b²), where a is half the span and b is the height.

Why is an elliptical arch better than a circular arch?

It provides a lower vertical rise for wide spans, making it ideal for rooms with low ceilings while still maintaining structural strength.

What do I need to draw an elliptical arch?

You need the span and rise measurements, two nails for the focal points, one nail for the apex, and a non-stretch string.

Arch Calculator - Free & Accurate Online Calculator

Arch Calculator

Enter the base length and the rise (height) of the arch to compute the radius, chord length, and focal distance needed to draw your arch accurately.

Length of the arch (base) — cm

Height of the arch (rise) — cm

Focus

To draw the arch’s outline:

Use a string of length equal to the desired arch’s length.
Fix the ends on focus 1 and focus 2.
With a pencil, move the string while drawing the arch’s perimeter.

Illustration showing how to draw an ellipse with a string.
© Omni Calculator

Elliptical Arch Calculator: Precise Geometry for Perfect Architectural Curves

Primary Goal	Input Metrics	Output	Why Use This?
Design and layout elliptical arches	Total Span ($S$), Arch Rise ($H$)	Focal Points & Plotting Coordinates	To achieve a structurally sound, low-profile curve that fits modern height constraints.

Understanding Elliptical Arches

An Elliptical Arch represents the top half of a horizontal ellipse. Unlike semi-circular arches, which require a rise equal to half the span, elliptical arches offer a “low-rise” profile. This makes them the superior choice for modern interior renovations where ceiling height is limited but a wide opening is desired.

Beyond aesthetics, the elliptical geometry is mathematically efficient. It distributes weight toward the vertical supports while minimizing the outward “thrust” (horizontal load) compared to a segmental arch. This reduces the risk of structural cracking in the surrounding masonry or drywall.

Who is this for?

Carpenters & Framers: To create accurate wooden templates (jigs) for doorway archways.
Architects & Interior Designers: To specify the exact curvature for custom window headers.
DIY Homeowners: To visually map out an arch on a wall before cutting.

The Logic Vault

The geometry of an elliptical arch is defined by its semi-major axis ($a$) and its semi-minor axis ($b$). To draw the arch accurately using the “string method,” you must identify the Focal Points ($F$).

Focal Distance Formula

$$f = \sqrt{a^2 – b^2}$$

Variable Breakdown

Name	Symbol	Unit	Description
Semi-Major Axis	$a$	mm / in	Half of the total span ($\frac{S}{2}$).
Semi-Minor Axis (Rise)	$b$	mm / in	The maximum height of the arch.
Focal Distance	$f$	mm / in	Distance from the center point to each focus.
Total Span	$S$	mm / in	The total horizontal width of the opening.

Step-by-Step Interactive Example

Let’s design an arch for a standard wide doorway with a 1200 mm span and a desired rise of 400 mm.

Determine Axis Values:
- $a = 1200 / 2 = \mathbf{600 \text{ mm}}$
- $b = \mathbf{400 \text{ mm}}$
Calculate Focal Distance ($f$):$$f = sqrt{600^2 – 400^2}$$$$f = sqrt{360,000 – 160,000}$$$$f = sqrt{200,000}$$$$f approx mathbf{447.2 text{ mm}}$$
Layout Instruction:Mark the center of your base. Measure 447.2 mm to the left and right of the center. These are your two focal points. Place a nail at each focus and at the peak (400 mm high), then use a non-stretch string to trace the perfect curve.

Information Gain: The “String Stretch” Error

The most common failure in physical arch construction is String Elasticity. Most builders use standard nylon string, which stretches under the tension of a pencil. Even a 2% stretch will distort the ellipse into an irregular “egg” shape, making the arch asymmetrical.

Expert Edge: Always use braided fishing line or thin picture-hanging wire. These materials have near-zero stretch (high tensile modulus), ensuring the $a + b$ constant of the ellipse remains mathematically perfect throughout the trace.

Strategic Insight by Shahzad Raja

Having reviewed thousands of architectural web tools over 14 years, I’ve noticed a critical SEO gap: most calculators ignore Springing Points. When framing an arch, you must account for the vertical “legs” before the curve begins. My specialized tip: Always subtract the thickness of your finishing material (like 12.5 mm drywall) from your calculated $a$ and $b$ values before cutting your structural timber. If you don’t, your finished arch will be smaller than the architectural plan intended.

Frequently Asked Questions

What is the “string method” for drawing an arch?

It is a technique where you tie a loop of non-stretch string around two nails at the focal points. When a pencil is placed in the loop and pulled taut to the apex, moving the pencil will naturally trace a perfect ellipse.

Why not just use a semi-circular arch?

A semi-circular arch is often too high for standard 2.4 m (8 ft) ceilings. For a 1.5 m wide opening, a semi-circle would require a 750 mm rise, which usually hits the ceiling. An elliptical arch can span the same 1.5 m with only a 300 mm rise.

How do I find the length of the actual curved arc?

The perimeter of an ellipse is complex and lacks a simple formula. However, Ramanujan’s Approximation is the gold standard for accuracy:

$$C \approx \pi [3(a+b) – \sqrt{(3a+b)(a+3b)}]$$

(Divide by 2 for the arch length).

Related Tools

Segmental Arch Calculator: For arches that are part of a circle rather than an ellipse.
Drywall Radius Tool: Estimate how much flexible drywall is needed for curved surfaces.
Stair Stringer Calculator: For planning the vertical supports leading up to an arched landing.