Radius, Chord and Diameter

Radius, Chord and Diameter — Moosa Academy

Three segments come up in almost every circle problem: the radius, the chord and the diameter. They are easy to confuse because the diameter is both a chord and twice a radius. Getting the definitions straight first makes every later circle theorem simpler.

Concept The three segments

Radius

r

— from the centre

C

to any point on the circle.

Chord — joins any two points on the circle; it need not pass through the centre.

Diameter

d

— a chord that does pass through the centre.

Theorem The relationships

All radii of one circle are equal. That is what makes it a circle.

The diameter is twice the radius:

d = 2r

, so

r = \dfrac{d}{2}

The diameter is the longest chord in the circle.

The one-way statement worth memorising: every diameter is a chord, but most chords are not diameters. Only the chords that happen to pass through the centre qualify.

Concept Why the diameter is the longest chord

Take any chord and join both of its endpoints to the centre. Those two segments are radii, so together they measure $2r$ . By the triangle inequality:

\text{chord} \leq r + r = 2r = d

Equality happens only when the triangle collapses into a straight line — that is, exactly when the chord passes through the centre. So no chord can beat the diameter, and only a diameter can match it.

Example Working between r and d

A circle has radius 7 cm. Find its diameter, circumference and area.

d = 2r = 2 \times 7 = 14 \text{ cm}

\text{Circumference} = 2\pi r = 2\pi \times 7 = 14\pi \approx 43.98 \text{ cm}

\text{Area} = \pi r^2 = \pi \times 49 = 49\pi \approx 153.94 \text{ cm}^2

Going the other way, a circle of diameter 20 cm has $r = 10$ cm. Always convert to the radius before using the area formula — squaring the diameter by mistake gives an answer four times too large.

Note The three compared

Segment	Starts at	Ends at	Through the centre?
Radius	The centre	The circle	Starts there
Chord	The circle	The circle	Not necessarily
Diameter	The circle	The circle	Always

Reading the last column top to bottom explains the whole family: the diameter is simply the chord that also behaves like two radii laid end to end.

Summary

The radius runs from the centre to the circle, and all radii of one circle are equal.
A chord joins two points on the circle and need not pass through the centre.
A diameter is a chord through the centre, and d = 2r.
The diameter is the longest chord, by the triangle inequality.
Every diameter is a chord, but not every chord is a diameter.
Circumference = 2πr and area = πr², both written in terms of the radius.