Spherometer: Parts, Working Principle, Least Count, Formula, and Complete Lab Guide

Of the three mandatory precision measuring instruments in the CBSE and ICSE Class 11 physics practical syllabus — the vernier caliper, the micrometer screw gauge, and the spherometer — the spherometer is perhaps the most elegant. Where the vernier caliper measures linear external dimensions (diameters, lengths) and the micrometer measures with even finer resolution (0.01 mm), the spherometer is the only instrument of the three that is designed specifically to measure the curvature of a surface: the radius of curvature of a spherical lens, mirror, or glass plate.

The spherometer achieves this remarkable capability through a beautifully simple geometric principle. By resting three equally-spaced legs on a spherical surface and measuring how much higher or lower the central screw stands relative to the plane of the three legs (a measurement called the sagitta), it uses a known geometric relationship to calculate the radius of the sphere to an extraordinary resolution of 0.001 cm (0.01 mm) — making it the most sensitive mechanical measuring instrument in the standard physics laboratory kit.

This guide covers the spherometer completely: its parts, working principle, least count derivation, the radius of curvature formula with a worked example, step-by-step reading procedure, types of spherometers, uses across optics and metrology, and guidance for school and college procurement. All spherometers described are manufactured by AJKANT Overseas in Ambala, Haryana — India’s scientific instrument capital.

1. What is a Spherometer?

A spherometer is a precision measuring instrument used to determine the radius of curvature of a spherical surface — such as a convex or concave lens face, a curved mirror, or any precisely ground spherical surface. The name derives from the Greek sphaira (sphere) and the Greek/Latin metron (measure): an instrument that measures spheres.

Unlike a ruler (which measures length directly) or a vernier caliper (which measures external or internal linear dimensions), the spherometer measures the departure from flatness of a curved surface. It does this by measuring the sagitta — the vertical distance between the flat plane defined by its three outer legs and the point where the central screw tip touches the spherical surface. Using the known geometry of the instrument (the distance between the legs) and the measured sagitta, the radius of curvature of the surface can be calculated directly using a simple algebraic formula.

In CBSE and ICSE Class 11 physics practicals, the spherometer is used in Experiment 1 to measure the radius of curvature of a given spherical surface (typically a convex lens or concave lens face), and this measurement is used to verify the lens-maker’s equation in subsequent experiments.

2. Parts of a Spherometer and Their Functions

A standard spherometer used in school and college physics laboratories has the following components:

3. Working Principle — The Screw and Sagitta Method

The spherometer works by combining two principles:

1. Screw Thread Principle: Like the micrometer, the spherometer uses a precision screw to convert rotational motion into precise linear displacement. One complete rotation of the disc advances the central screw by one pitch (0.5 mm). The disc scale divides each rotation into 100 parts (0.005 mm per division), giving the resolution of the instrument.
2. Sagitta Geometry: The sagitta (h) is the vertical distance between the flat plane defined by the three outer legs and the point where the central screw touches the spherical surface. If the surface is convex, the screw must be lowered below the plane of the legs to touch it (h is positive). If the surface is concave, the screw must be raised above the plane of the legs (h is negative). The absolute value of h is the sagitta of the spherical arc between the three leg-contact points.

Once the sagitta h and the mean leg separation distance l are known, the radius of curvature R of the spherical surface is computed from the spherometer formula (derived from the geometry of a sphere and its chord).

4. Spherometer Least Count — Formula and Calculation

The least count of the spherometer is the minimum vertical displacement it can reliably resolve. It is calculated exactly as for any screw-gauge instrument:

This means the spherometer can resolve a vertical displacement of just 5 micrometres (0.005 mm) — making it the most sensitive standard mechanical instrument in the school physics laboratory. This resolution is twice as fine as the standard micrometer screw gauge (0.01 mm) and ten times finer than the standard vernier caliper (0.02 mm).

5. Radius of Curvature Formula — Derivation and Worked Example

The radius of curvature formula for the spherometer is derived from the geometry of a sphere, its chord, and the sagitta. For a sphere of radius R, where three points on the sphere lie at the tips of the outer legs (separated by mean distance l from each other) and the sagitta of the arc is h:

Worked Example — Finding Radius of Curvature of a Convex Lens

6. How to Use a Spherometer: Step-by-Step Procedure

1. Measure the Mean Distance Between Outer Legs (l)

   Place the spherometer on a sheet of paper and press it gently to mark the three leg positions as dots. Using a ruler or vernier caliper, measure the distance between each pair of dots (l₁₂, l₂₃, l₁₃). Calculate the mean: l = (l₁₂ + l₂₃ + l₁₃) / 3. This value of l is fixed for a given spherometer and need only be measured once.
2. Determine the Least Count

   Note the pitch (0.5 mm for standard instruments) and count the number of disc divisions (100 for standard instruments). Calculate: LC = 0.5 mm / 100 = 0.005 mm. Record this in your practical notebook before taking any measurements.
3. Set the Zero Reading on the Flat Glass Plate

   Place the flat reference glass plate on a flat surface. Set the spherometer on the plate. Rotate the disc until the central screw tip just touches the plate surface (all four contact points — three legs + screw tip — rest on the plate). Note the disc division aligned with the vertical scale datum. This is your zero reading. Note any zero error (same concept as for the micrometer).
4. Place the Spherometer on the Curved Surface

   Remove the glass plate. Place the spherometer on the given convex or concave spherical surface (e.g., a convex lens face). The three outer legs will rest on the sphere’s surface. Now rotate the disc to advance or retract the central screw until its tip also just touches the spherical surface. Note: for a convex surface, the screw moves down (positive sagitta); for a concave surface, the screw moves up (negative sagitta).
5. Read the Main Scale and Disc Scale

   Main Scale Reading (MSR): Count the number of complete 0.5 mm divisions visible on the vertical scale above (or below) the disc. Circular Scale Reading (CSR): Read the disc division aligned with the datum line on the vertical scale. Total sagitta h = MSR + (CSR × LC). Subtract zero error if present.
6. Repeat for Multiple Readings

   Lift and re-place the spherometer at least three different positions on the spherical surface. Record each reading independently and calculate the mean sagitta h̄. This eliminates positioning errors and gives a more reliable measurement.
7. Calculate Radius of Curvature

   Substitute the mean values of l and h into the formula: R = l² / (6h) + h/2. Express the result in the same units used for l and h (cm or mm). Compare with the nominal focal length of the lens if known.

7. Types of Spherometers

8. Spherometer vs. Vernier Caliper vs. Micrometer — Which Measures What?

9. Spherometer Uses in Laboratory and Industry

10. Frequently Asked Questions (FAQ)