Optics Sag Calculator: Formula & Example

An optics sag calculator works out how much the center of a circular optical element (like a mirror or lens blank) drops below the plane of its outer edge when it is supported at the rim. You enter the diameter (or radius) and the radius of curvature of the curved surface; the calculator returns the sagitta (usually called “sag”) in the same length unit you used.

Formula

The sagitta of a spherical surface is given by:

s = R − √(R² − (d/2)²)

• s = sagitta (mm, inches, or any consistent unit)
• R = radius of curvature of the surface (same unit as s)
• d = diameter of the mirror or lens (same unit)

For a mirror, the focal length f = R / 2, so the focal ratio is f/D = R / (2d). The sag formula is exact for a sphere; it does not assume a parabola. For a parabolic mirror, the sag at the center is the same to first order, but the exact surface shape differs.

Worked Example

Using the defaults: diameter d = 200 mm, radius of curvature R = 2000 mm.

1. Half the diameter: d/2 = 100 mm
2. Square it: (100)² = 10,000 mm²
3. Square R: 2000² = 4,000,000 mm²
4. Subtract: 4,000,000 − 10,000 = 3,990,000 mm²
5. Square root: √3,990,000 ≈ 1997.498 mm
6. Subtract from R: 2000 − 1997.498 = 2.502 mm

The calculator gives 2.502 mm (rounded to 3 decimals). The focal ratio is (2000 / 2) / 200 = 5.00, so this is an f/5 mirror.

Why Sag Matters in Optics

Sagitta is the depth you need to grind or polish into a glass blank to get a target focal length. Amateur telescope makers use it constantly:

• To choose a grinding tool or lap size
• To estimate how much glass must be removed
• To check progress with a spherometer (which measures sag directly)
• To design a mirror cell or support points

For lenses, sag is used when calculating edge thickness from center thickness, or when designing a convex surface on a plano-convex lens. Optical designers also use sag to compute surface sagitta for ray tracing.

How to Use the Calculator Correctly

Enter the diameter of the optical surface, not the overall blank if it is larger than the polished area. The radius of curvature must be measured along the optical axis, not along the surface. Use the same unit for both inputs – mixing mm and inches is the most common error.

The result is the depth at the center relative to a flat plane through the edge. If you need the sag for only half a surface (e.g., a off-axis segment), use the chord length of that segment as the diameter.

Common Mistakes and Misconceptions

• Using the focal length instead of R: For a mirror, R = 2 × focal length. If you enter the focal length directly, the sag will be roughly half the correct value.
• Confusing sag with “depth of curve” measured by a spherometer: A spherometer measures sag over its own ring diameter, not the mirror’s full diameter. You must use the spherometer’s ball spacing in the formula, not the mirror diameter.
• Assuming the formula works for a parabola: The exact sag of a parabola is s = d² / (16 f) = d² / (8R). For a sphere, the sag is slightly larger. The difference is tiny for slow mirrors (f/10) but becomes significant for fast mirrors (f/3).
• Rounding too early: If you round R² and the square root to fewer digits, you can get a sag error of several microns. Keep at least 4 significant figures in intermediate steps.

Limits of the Formula

The formula assumes a perfect spherical surface. Real mirrors are often parabolic, and the sag for a parabola is:

s_parabola = d² / (16 f) = d² / (8R)

For the example above, that gives 200² / (8×2000) = 40,000 / 16,000 = 2.500 mm – only 0.002 mm less than the spherical value. For an f/3 mirror (R = 1200 mm, d = 400 mm), the difference is about 0.04 mm, which matters when figuring.

The formula also fails for very deep curves where R is close to d/2 – that would imply a hemisphere or deeper, which is rare in ordinary optics. In that regime, the surface is no longer well described by a simple sag.

Practical Tips for Telescope Makers

• Use a spherometer with a known ring radius to measure sag during grinding. Convert its reading to the mirror’s full sag using the same formula.
• When polishing a mirror, aim for the sag that gives your target focal length. You can check with a knife-edge test or a Foucault tester.
• For a mirror support cell, the sag tells you how much the mirror will flex if supported at the edge. The formula for a flat plate under its own weight is different – that involves plate thickness and material properties.
• If you buy a mirror blank, the sag is often given in the spec sheet. Use it to verify the radius of curvature.

Frequently Asked Questions

What is sagitta in simple terms?

Sagitta is the depth of the curve at the center of an optical surface, measured from a flat plane that touches the edge. It is the “drop” you would feel if you placed a straightedge across the mirror and measured the gap at the center.

Why is sag important for a telescope mirror?

The sag determines the focal length. For a spherical mirror, focal length = R/2. For a parabolic mirror, the sag is nearly the same, but the exact shape is different. If the sag is wrong, the mirror will have the wrong focal length and the telescope won’t focus correctly.

Can I use this calculator for a lens?

Yes, for a single spherical surface of a lens. For a biconvex lens, you need to calculate the sag for each surface separately and add them to get the total center thickness minus edge thickness. The lensmaker’s equation uses the radii of curvature, not the sag directly.

What is the difference between sag and “depth of curve”?

They are the same thing. “Depth of curve” is the common term used by telescope makers. Sagitta is the technical term. A spherometer measures the depth over its own ring, so you must convert to the mirror’s full diameter.

Should I use the spherical or parabolic sag formula?

If you are grinding a spherical mirror (for a beginner’s first telescope), use the spherical formula. If you are making a parabolic mirror, use the parabolic formula for the final figure, but the difference is tiny for slow mirrors. For fast mirrors (f/4 or faster), use the parabolic formula.

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in d | Diameter | mm | 200 | 0
in R | Radius of curvature | mm | 2000 | 0
out sag | Sagitta | mm | R - sqrt(R^2 - (d/2)^2) | 3
out f_ratio | Focal ratio (f/D) | - | (R / 2) / d | 2