How this instrument works
A circle's diameter is its longest possible chord: a straight line through the center connecting two points on the circle, equal in length to two radii laid end to end. That is why d = 2r holds by construction rather than by measurement — walk from the center out to the edge (that distance is r), then keep walking the same distance across to the far edge, and you have traced the diameter. No approximation, no transcendental constant, just doubling.
This is the one circle relationship that never touches π. Circumference and area both scale by that irrational constant, but the radius-diameter pair is pure arithmetic, so d = 2r is exact to as many digits as r itself carries. That exactness matters in practice: a drill bit, a pipe fitting, or a wheel hub is almost always labeled by its diameter, while the geometry and physics formulas built on that part — area, moment of inertia, stress — want the radius instead. Halving when you should double, or the reverse, is a genuinely common fitting mistake.
The definition carries a payoff beyond arithmetic. Thales' theorem says that if you pick any diameter of a circle and any third point on the circle, the triangle they form has a right angle sitting exactly opposite that diameter — a fact the ancient Greeks used to construct right angles with nothing but a compass and a straightedge. At the far edge of the scale, a radius of zero collapses the diameter to zero as well: the circle shrinks to a single point, the smallest case this formula can describe.
- Enter your circle's measurement into the Radius field — any unit works, so long as you read the result in that same unit.
- Diameter updates beside it automatically, computed as exactly twice the value you entered.
- Use that Diameter figure wherever a spec sheet, bore, or drill chart calls for the width straight across, not the distance from the center.
- To go the other direction — a known diameter, and a radius you need — halve the Diameter figure yourself; the relationship runs both ways, r = d ⁄ 2.
Worked example — a radius-5 wheel hub
A wheel hub is machined with a radius of exactly 5 cm, so its diameter is d = 2 × 5 = 10.0 cm — precisely, not approximately, since no π enters a radius-to-diameter conversion. That 10.0 cm figure is the number to check against the bearing bore stamped on the parts drawing, because bore sizes are always given as a diameter, never a radius.
Halve that same figure and the radius returns exactly: 10.0 ⁄ 2 = 5 cm, with nothing lost in the round trip. Compare a smaller hub with a radius of 0.5 cm, which comes out to a diameter of exactly 1 cm — a convenient unit-diameter reference size that turns up throughout geometry problems.
Questions
What is the difference between radius and diameter?
Radius is the distance from the center of a circle to its edge; diameter is the distance all the way across, straight through the center. Diameter is always exactly twice the radius, d = 2r, because it is built from two radii placed end to end along the same line.
How do you convert a diameter back to a radius?
Divide by two: r = d ⁄ 2. It is the same relationship run backwards, and it costs nothing in accuracy — a diameter of 10 cm gives a radius of exactly 5 cm, with no rounding beyond whatever precision the measured diameter already carried.
Does finding a diameter from a radius involve π?
No. π enters only when a circle's curved boundary is involved, as in circumference (2πr) or area (πr²). The radius-diameter pair is a straight-line relationship, two radii laid end to end, so d = 2r is exact arithmetic rather than an approximation of an irrational constant.
Why do parts and hardware get labeled by diameter instead of radius?
Diameter is what a caliper or tape measure reads directly across a round part, while radius requires locating a center point first, which is often invisible on a finished pipe, bolt, or wheel. Drill bits, pipes, and bearings are labeled by diameter for that reason, even though downstream formulas usually want the radius.
What is Thales' theorem, and why does it single out the diameter?
It states that any triangle drawn with one side as a diameter, and its opposite vertex anywhere else on the circle, has a right angle at that vertex. The result holds only for diameters and fails for any shorter chord, which is what makes the diameter more than just the widest measurement across a circle.
What happens to the diameter when the radius is zero?
It becomes zero too: d = 2 × 0 = 0. A zero-radius circle is a degenerate case, a single point rather than a curve, and it marks the lower boundary of the formula — every larger radius produces a proportionally larger, well-defined diameter.