SAG Calculator Pro (USA) – Free Online Sagitta Calculator for Optics, Mirrors & Engineering
USA Edition

SAG Calculator Pro

Precision sagitta calculator for optics, telescope mirrors, glass fabrication, and engineering. Free, accurate, and built for USA professionals.

Input Parameters

Visual Diagram

Enter values and click Calculate to see the interactive diagram

Step-by-Step Solution

Common USA Telescope Mirror SAG Values

Click any row to auto-fill the calculator with those values.

Diameterf-ratioFocal LengthRadius (R)SAG (exact)
4.25 inf/834.00 in68.00 in0.0332 inClick
6.00 inf/848.00 in96.00 in0.0469 inClick
8.00 inf/648.00 in96.00 in0.0833 inClick
10.00 inf/550.00 in100.00 in0.1250 inClick
12.50 inf/562.50 in125.00 in0.1564 inClick
16.00 inf/4.572.00 in144.00 in0.2222 inClick
20.00 inf/480.00 in160.00 in0.3125 inClick
25.00 inf/3.895.00 in190.00 in0.4118 inClick

How SAG Calculation Works

The sagitta (SAG) is a fundamental geometric measurement that defines the depth of a circular arc. It is the perpendicular distance from the midpoint of a chord to the arc's midpoint. Understanding this measurement is essential in precision optics, telescope mirror fabrication, architectural engineering, and CNC machining.

Core Formulas

SAG = R − √(R² − (D/2)²)
Exact formula: R = radius of curvature, D = chord diameter
R = D² / (8 × SAG) + SAG / 2
Inverse formula: find radius when SAG and diameter are known
SAG ≈ D² / (8 × R)
Approximate formula (accurate when SAG is small relative to R)
R = 2 × f
Mirror relationship: radius of curvature equals twice the focal length

Additional Calculations Provided

  • Arc Length: L = 2 × R × arcsin(D / (2R)) — the actual curved distance along the arc
  • Circular Segment Area: A = R² × arcsin(D/(2R)) − (D/2) × √(R² − (D/2)²)
  • Spherical Cap Surface Area: SA = 2π × R × SAG
  • Spherical Cap Volume: V = π × SAG² × (3R − SAG) / 3
  • f-ratio: f/D — critical for telescope mirror specification

When to Use Exact vs. Approximate Formula

The approximate formula SAG ≈ D²/(8R) introduces error that increases with the SAG-to-radius ratio. For typical telescope mirrors (f/5 and slower), the approximation error is under 0.1%. For fast mirrors (f/4 and faster) or deep curves in industrial applications, always use the exact formula. This calculator always uses the exact formula and displays both results for comparison.

Real-World Applications

Telescope Mirror Making (ATM)

Amateur telescope makers use SAG to verify the correct curvature during grinding and figuring. The sagitta determines whether the mirror will achieve the desired focal length and is measured with a spherometer or straightedge-and-feeler-gauge setup.

Optical Lens Manufacturing

Precision lenses for cameras, microscopes, and laser systems require exact sagitta specifications. CNC optical generators use SAG values to program curvature cuts on glass blanks to tolerances of ±0.0001 inches.

Architectural Arches & Domes

Structural engineers calculate sagitta to determine the rise of arches, vaults, and domes. This affects load distribution, material quantities, and aesthetic proportions in bridges and buildings per ASCE standards.

Sheet Metal & HVAC Ductwork

Curved sheet metal panels for HVAC ducts, aircraft skins, and automotive body panels require sagitta calculations to set press brake depths and roller positions accurately.

Concrete & Road Engineering

Crown and superelevation in road design, as well as curved concrete formwork, rely on sagitta to achieve proper drainage slopes and structural curves per AASHTO guidelines.

Pipeline & Pressure Vessel Design

ASME Boiler and Pressure Vessel Code (BPVC) calculations for cylindrical shell segments and domed heads use sagitta-based geometry for stress analysis and material specification.

Frequently Asked Questions

SAG, also called sagitta (Latin for "arrow"), is the distance from the midpoint of a chord to the midpoint of the arc it subtends. In optics, it represents the depth of a curved surface—such as a lens or mirror—measured from the edge (chord) to the center (vertex). It is essential for specifying the curvature of optical elements, telescope mirrors, architectural arches, and precision-machined parts.
The standard SAG formula is: SAG = R − √(R² − (D/2)²), where R is the radius of curvature and D is the chord diameter. For small sagitta relative to the radius, the approximate formula SAG ≈ D² / (8R) is often used. The inverse formula to find radius from sagitta is: R = D²/(8×SAG) + SAG/2.
You can use any consistent unit system. In the USA, inches are most common for optics and machining (e.g., a 10-inch mirror with 0.125-inch sagitta). Metric units (millimeters) are used in scientific contexts. This calculator supports inches, feet, millimeters, centimeters, and meters with automatic conversion.
For a spherical mirror, the radius of curvature (R) equals twice the focal length (f): R = 2f. Once you know R and the mirror diameter (D), you can calculate the SAG using the standard sagitta formula. For example, an f/8 mirror with a 10-inch diameter has a focal length of 80 inches, a radius of 160 inches, and a sagitta of approximately 0.078 inches.
SAG and chord height refer to the same measurement—the perpendicular distance from the midpoint of a chord to the arc. The terms are used interchangeably in optics, engineering, and geometry. "Sagitta" is the Latin term commonly used in optics, while "chord height" or "rise" is more common in structural engineering and architecture.