Tony Foale has described a radius gauge...
based on the principle of measuring a chord and its related sagitta and then using these values to compute the radius of the measured curve.
The same principle can be applied to measure the diameter of circular objects whose diameter exceeds the capacity of your calipers. (Capacity here can refer to: calipers whose jaws do not open far enough to encompass the diameter to be measured or calipers whose jaws are too short to reach the diameter of the object.)
The first photo shows a too small caliper attempting to measure the diameter of an object. The caliper's jaws are too short to reach the diameter so it ends up measuring the length of some chord of the circle formed by the object. Note how the object is touching the beam of the caliper; it's pushed as far into the gape of the caliper as it will go.
This diagram will help to understand the mathematical notation...
The tips of the caliper jaws are at locations B and D. The line BD is the chord and that is what the calipers are measuring. The sagitta, EC (marked S in the diagram) is the distance from the chord to the circle's circumference. It should be obvious that that is the length of the caliper's jaws. Just to be clear, the following picture shows what you need to measure to find the length of your caliper's jaws...
(but use calipers not a scale; the picture is meant to show what to measure, not how to measure it.)
Now, if you know the length of the chord and the sagitta, you can compute the radius of the curve from a simple formula...
r = (c^2 + 4 * s^2) / (8 * s)
d = 2 * r = (c^2 + 4 * s^2) / (4 * s)
c = chord
s = sagitta
r = radius
d = diameter
How well does it work ? Well, of course, that depends on your instruments and how carefully you measure.
For the bench block pictured, I made the following measurements...
c = 2.859
s = 1.159
and computed a diameter of 2.922. I then used my height gauge to measure the diameter (the jaws of my larger calipers are too short) and got a diameter of 2.924. YMMV.