What Are Chords and Diameters?
A chord is a straight line segment whose two endpoints both lie on a circle. A diameter is a chord that passes through the center of the circle. So every diameter is a chord, but only the chords running through the center earn the name diameter.
Line up the parts of a circle and the relationship is clean. The radius $r$ runs from the center to the edge. The diameter $d$ runs edge-to-edge through the center, so it is exactly two radii laid end to end: $d = 2r$. A chord is any edge-to-edge segment, whether or not it visits the center.
One more neighbour is worth naming so it does not get confused later: a secant is the full straight line that a chord sits on, extended past the circle in both directions. A chord is the piece inside; the secant is the whole line.
The Instrument That Cannot Tune Without a Chord
Stretch a string across the round soundhole of a guitar and pluck it: where you stretch it changes the note, and the longest string you can possibly stretch runs straight through the middle. That longest string is the diameter. Every shorter string is a chord. The whole geometry of chords and diameters is hiding in that one observation, the closer a chord runs to the center, the longer it gets, and the winner passes through the center itself.
Why Is the Diameter the Longest Chord?
Reach for the reason and it is tempting to say "because it passes through the center." That is true, but it is really a definition restating itself, it does not prove the length. The clean proof uses the triangle inequality.
Take any chord $AB$ that does not pass through the center $O$. Join $OA$ and $OB$; both are radii, so $OA = OB = r$. Now $A$, $O$, $B$ form a triangle, and the triangle inequality says one side is always shorter than the sum of the other two:
$$AB < OA + OB = r + r = 2r$$
So every chord that misses the center is strictly shorter than $2r$. A diameter, running edge-to-edge through $O$, measures exactly $2r$. Nothing beats it, and only the diameter ties the maximum. This is the reasoning Euclid gives, and it is why "the diameter is the longest chord" is a theorem, not just a description.
What Are the Properties of Chords and Diameters?
The behaviour of chords in a circle follows a handful of properties that every circle problem leans on.
The diameter is the longest chord, with length $2r$; every other chord is shorter.
Equal chords are equidistant from the center. Two chords the same length sit the same perpendicular distance from the center, and the converse holds too.
The nearer the center, the longer the chord. As a chord's distance from the center shrinks, its length grows, reaching maximum ($2r$) when the distance hits zero.
A perpendicular from the center bisects the chord. The line from the center meeting a chord at a right angle cuts it into two equal halves, the basis of the perpendicular bisector of a chord passing through the center.
Equal chords subtend equal angles at the center. Chords of the same length open the same central angle.
These are not five separate facts to memorise so much as five views of one idea: distance from the center controls everything about a chord.
How Do You Find the Length of a Chord?
There are two standard routes, and which you use depends on what you are given.
Given the radius and the chord's distance from the center. Drop a perpendicular from the center $O$ to the chord, meeting it at $M$. Because that perpendicular bisects the chord, $M$ is the midpoint, and $OM = d$. Now $OMA$ is a right triangle with hypotenuse $r$ and legs $d$ and $AM$. By Pythagoras:
$$AM = \sqrt{r^2 - d^2}$$
The full chord is twice the half-chord:
$$\text{chord length} = 2\sqrt{r^2 - d^2}$$
Given the radius and the central angle $\theta$. The chord and the two radii form an isosceles triangle, and trigonometry gives:
$$\text{chord length} = 2r\sin\left(\frac{\theta}{2}\right)$$
Both formulas agree with the headline fact: push $d \to 0$ (or $\theta \to 180°$) and the length climbs to $2r$, the diameter.
Examples of Chords and Diameters
Example 1
A circle has radius $8$ cm. How long is its diameter?
The diameter is twice the radius.
$$d = 2r = 2 \times 8 = 16 \text{ cm}$$
Final answer: the diameter is $16$ cm.
Example 2
A chord lies $3$ cm from the center of a circle of radius $5$ cm. Find the chord's length.
Wrong attempt. A common first move is to compute the half-chord and call it the answer: $\sqrt{5^2 - 3^2} = \sqrt{25 - 9} = \sqrt{16} = 4$, so the chord is $4$ cm. But $4$ cm is only the distance from the midpoint to one endpoint, half the chord. The perpendicular from the center lands in the middle of the chord, so we have measured just one side of it.
Correct. Double the half-chord to span the whole thing:
$$\text{chord} = 2\sqrt{r^2 - d^2} = 2\sqrt{16} = 2 \times 4 = 8 \text{ cm}$$
Final answer: the chord is $8$ cm long.
Example 3
Is a diameter a chord?
Yes. A chord is any segment joining two points on the circle, and a diameter does exactly that, it just happens to pass through the center as well. Every diameter is a chord; not every chord is a diameter.
Final answer: yes, a diameter is a chord (the longest one).
Example 4
A chord of a circle is $24$ cm long and sits $5$ cm from the center. Find the radius.
Half the chord is $12$ cm, and it forms a right triangle with the distance $5$ cm and the radius as hypotenuse:
$$r = \sqrt{12^2 + 5^2} = \sqrt{144 + 25} = \sqrt{169} = 13 \text{ cm}$$
Final answer: the radius is $13$ cm.
Example 5
Two chords of a circle are each $10$ cm long. What can you say about their distances from the center?
By the equal-chords property, chords of the same length are equidistant from the center. So both chords lie the same perpendicular distance from the center.
Final answer: the two chords are equidistant from the center.
Example 6
A chord subtends a central angle of $60°$ in a circle of radius $6$ cm. Find the chord's length.
Use the central-angle formula:
$$\text{chord} = 2r\sin\left(\frac{\theta}{2}\right) = 2 \times 6 \times \sin 30° = 12 \times \tfrac{1}{2} = 6 \text{ cm}$$
Final answer: the chord is $6$ cm long.
Where Do Chords and Diameters Show Up?
Chords and diameters are how engineers pin down circles they cannot see the center of. A broken circular gear, an arc of an old stone bridge, a curved architectural fragment, measure a chord and its distance from where the center would be, and $2\sqrt{r^2 - d^2}$ hands you the radius of the original whole.
Reconstructing broken arcs. Archaeologists and restorers find the original radius of a shattered wheel or plate from a single surviving chord.
Navigation and surveying. A "chord distance" is the straight-line shortcut across a curved path; comparing it to the arc tells surveyors how much the curve bends.
Engineering tolerances. Machinists check whether a circular part is true by measuring chords at fixed depths.
The diameter as a size label. Pipes, bolts, lenses, and wheels are all specified by diameter, the single longest chord that captures how big the circle is. The standing history of this reasoning is catalogued at Wolfram MathWorld.
Where Do Students Trip Up on Chords and Diameters?
Mistake 1: Reporting the half-chord as the whole chord
Where it slips in: any chord-length problem that goes through the perpendicular-from-center right triangle.
Don't do this: stopping at $\sqrt{r^2 - d^2}$ and calling it the chord.
The correct way: that square root is only half the chord, because the perpendicular bisects it. Double it: chord $= 2\sqrt{r^2 - d^2}$. The perpendicular from the center always lands at the midpoint, and forgetting the factor of two is the single most common error on this topic.
Mistake 2: Assuming every chord through a nice-looking point is a diameter
Where it slips in: diagrams where a chord passes close to, but not through, the center.
Don't do this: treating a chord as length $2r$ just because it looks long.
The correct way: a chord is a diameter only if it passes through the exact center. Off by a whisker, and its length is strictly less than $2r$. The rusher who eyeballs "that's basically the middle" pays for it, check that the segment actually contains the center before you call it a diameter.
Mistake 3: Confusing the chord with the arc
Where it slips in: length questions that mix the straight chord with the curved arc between the same two endpoints.
Don't do this: using the chord formula when the question wants the curved distance (or vice versa).
The correct way: the chord is the straight segment; the arc is the curved path. They share endpoints but are different lengths, and the arc is always longer. Name which one the problem asks for before reaching for a formula.
Conclusion
A chord joins two points on a circle; a diameter is a chord through the center, so the diameter is the longest chord at length $2r$.
The diameter wins by the triangle inequality: any off-center chord is shorter than the two radii that reach its endpoints.
Chord length from the center distance is $2\sqrt{r^2 - d^2}$, remember to double the half-chord.
Equal chords sit equidistant from the center, and chords grow longer the closer they run to it.
A chord (straight) is not the arc (curved) between the same two points; keep them apart.
To take chords and diameters further with a teacher, explore Bhanzu's geometry tutor, a middle school math tutor, or live math classes online.
Practice These to Solidify Your Understanding
Try these three, drawing the perpendicular-from-center triangle each time: (1) find the diameter of a circle with radius $9.5$ cm; (2) a chord $16$ cm long lies $6$ cm from the center, find the radius; (3) find the length of a chord $8$ cm from the center in a circle of radius $10$ cm. If the factor-of-two step trips you, return to How Do You Find the Length of a Chord? above. Want a live Bhanzu trainer to work through more chord problems? Book a free demo class.
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