What You Will Learn

• How sag relates to radius of curvature
• Limitations for shallow curves
• Calibration requirements

A spherometer is a precision gauge with three feet arranged in a circle and a central probe connected to a dial or digital indicator. The three feet define a stable reference plane. When the probe touches an optical surface, the indicator reports the sag, how far the surface at the probe is above or below the feet’s plane. By convention in many shops, positive sag indicates a convex surface under the probe and negative sag indicates concave. Because a spherical surface has a constant radius of curvature, a single sag reading, combined with the tool’s leg-circle size, lets you compute the radius of curvature.

Spherometers are most effective on short radii (deeper curves) where sag is larger and easier to measure. On very long, shallow curves, the sag can be just a few micrometers. At those tiny values, dust, vibration, and indicator resolution can dominate the reading and make results noisy.

Parts of a Spherometer

A typical shop spherometer includes a rigid ring with three hardened feet that form a circle, a center probe that moves relative to that plane, and an indicator that displays sag in millimeters or micrometers. The leg-circle size (often engraved on the frame) may be given as a diameter; for calculations you’ll convert that to a radius. A reference flat or a stack of gauge blocks is used to set the zero before measuring optics.

How to Use It (Step-by-Step)

First, make the measurement environment stable: set the part on a clean, flat, lint-free pad on a sturdy surface away from fans or vibration.

• Clean & inspect. Lightly wipe the spherometer feet and the optic with approved wipes. Even a small particle under a foot can add micrometers to the sag and skew the radius.
• Zero on a reference flat. Place all three feet and the probe on a clean reference flat (or wringed gauge blocks) and set the indicator to zero. This establishes that a perfectly flat surface reads 0 sag.
• Place on the optic. Lower the tool gently onto the surface. Avoid rocking; let the probe contact naturally. If the probe has spring preload, lower until the preload is engaged consistently.
• Read the sag. Record the indicator value and its sign. Note the measurement units (mm or µm) and the spherometer’s leg-circle size.
• Compute the radius. Use the formulas in the box below. If the leg circle marking is a diameter, divide by two to get the radius for the formula.

Choosing Leg-Circle Size

A larger leg-circle reduces sensitivity on deep curves but increases contact stability and averages local surface errors. A smaller circle increases sensitivity (larger sag for the same radius) but is more sensitive to local pits/scratches and rocking. Match the leg-circle to your expected radius: smaller for short radii, larger for long radii.

Sign Convention & Interpreting Readings

When the surface bulges toward the probe (convex), the probe sits higher relative to the feet, positive sag in the common convention. When the surface dips away (concave), the probe sits lower, negative sag. Always confirm your shop’s sign convention; if your indicator is flipped, you may need to invert the sign in the radius formula.

Use consistent units (mm and µm). Keep the tool vertical and avoid pressing on the frame. Average several readings if the value drifts. On very shallow curves (tiny sag), small errors in zero or dust can swing the computed radius by a lot, consider a different method (e.g., test plate or interferometer) if the required accuracy is high.

• Spherometers assume the measured area is spherical. On aspheres or near edges, readings can mislead.
• The smaller the sag compared to the leg-circle radius, the more sensitive the radius calculation is to noise.
• Always document the leg-circle size, sag value, sign, temperature (if critical), and how you zeroed.

Calibration Requirements

Zero on a certified flat before each session. Verify the leg-circle size (radius or diameter) against the engraving or documentation. If available, check against a reference sphere of known radius to confirm your sign convention and computation.

Lesson 6 · Spherometer Trainer

Drag the tool so the feet sit on the lens. Visuals are ×5 exaggerated for teaching; numbers remain exact.

Feet circle radius (a): 15.00 mm Surface: Convex

Indicator & Result

Sag0.00000 mm

Radius R∞

Zero on Flat Clear Zero Reset Hint

Set Up the Measurement

Leg circle diameter (2a)

40.00 mm

Surface radius |R|

120.0 mm

Convex (+s)

?

Convex surface bulges outward; sag is positive.

Concave (−s)

?

Concave surface curves inward; sag is negative.

Flat

?

Flat surface → sag ≈ 0; radius tends to infinity.

How it works

The three feet define a plane (a circle of radius a). The probe moves relative to that plane and reads the sag s. For a spherical surface:

s = R − √(R² − a²)  and  R = (a² + s²) / (2s)

Shallow curves → tiny s → harder to measure.

Use the range sliders to change 2a and |R|. Zero simulates calibration on a flat. Reset recenters the tool.