What Is a Chord? Definition, Properties & Examples

What Exactly Is a Chord?

A chord is the line segment connecting any two distinct points on a curve. For a circle, both endpoints lie on the circumference, and the chord lies entirely inside the circle.

Three facts define how chords behave. Each one comes up in nearly every circle problem you will meet.

• The diameter is the longest chord. Any chord through the centre is a diameter; no chord can be longer than the diameter.

• A perpendicular from the centre bisects the chord. Drop a line from the centre at a right angle to a chord, and it cuts the chord into two equal halves. This is the most-used chord property in geometry.

• Equal chords are equidistant from the centre. Two chords of the same length sit the same distance from the centre — and conversely, chords closer to the centre are longer.

You will also meet a chord wherever a straight line crosses a curve, not only a circle: the segment a secant line cuts off inside an ellipse or a parabola is a chord too. The circle is just where the idea is cleanest.

How Do You Find the Length of a Chord?

There are two standard chord-length formulas, and which one you reach for depends on what the problem gives you.

Given the radius $r$ and the perpendicular distance $d$ from the centre to the chord:

$$\text{Chord length} = 2\sqrt{r^2 - d^2}.$$

This drops straight out of the Pythagorean theorem: the radius, the half-chord, and the distance $d$ form a right triangle.

Given the radius $r$ and the central angle $\theta$ the chord subtends:

$$\text{Chord length} = 2r \sin!\left(\frac{\theta}{2}\right).$$

Both formulas describe the same segment; they just start from different given information.

Examples of a Chord

Example 1

A circle has radius $13$ cm. A chord lies $5$ cm from the centre. Find its length.

Use $2\sqrt{r^2 - d^2}$ with $r = 13$ and $d = 5$:

$$2\sqrt{13^2 - 5^2} = 2\sqrt{169 - 25} = 2\sqrt{144} = 2 \times 12 = 24 \text{ cm}.$$

Final answer: $24$ cm.

Example 2

A chord of a circle of radius $10$ cm is $12$ cm long. How far is it from the centre?

Wrong attempt. A student takes the full chord, $12$, and plugs it in as the leg of the right triangle: $d = \sqrt{10^2 - 12^2} = \sqrt{100 - 144} = \sqrt{-44}$ — a square root of a negative number, which is the signal the setup is wrong.

Where it broke. The perpendicular from the centre bisects the chord, so the right triangle uses the half-chord, $6$, not the full $12$.

Correct. $d = \sqrt{r^2 - (\text{half-chord})^2} = \sqrt{10^2 - 6^2} = \sqrt{100 - 36} = \sqrt{64} = 8 \text{ cm}.$

Final answer: $8$ cm.

Example 3

A chord subtends a central angle of $60°$ in a circle of radius $8$ cm. Find its length.

Use $2r\sin(\theta/2)$ with $r = 8$ and $\theta = 60°$:

$$2 \times 8 \times \sin(30°) = 16 \times \tfrac{1}{2} = 8 \text{ cm}.$$

Final answer: $8$ cm.

Example 4

Two chords of a circle are equal in length. Chord $PQ$ is $7$ cm from the centre. What can you say about the distance of the other chord, $RS$, from the centre?

Equal chords are equidistant from the centre, so $RS$ is also $7$ cm from the centre.

Final answer: $7$ cm.

Example 5

A diameter is $26$ cm. Find the radius, then the length of a chord lying $10$ cm from the centre.

The radius is half the diameter: $r = 13$ cm. Then:

$$\text{Chord} = 2\sqrt{13^2 - 10^2} = 2\sqrt{169 - 100} = 2\sqrt{69} \approx 16.6 \text{ cm}.$$

Final answer: about $16.6$ cm.

Example 6

Find the length of the longest possible chord in a circle of radius $9$ cm.

The longest chord is the diameter, which is twice the radius:

$$\text{Longest chord} = 2r = 2 \times 9 = 18 \text{ cm}.$$

Check against the distance formula: the diameter passes through the centre, so $d = 0$, giving $2\sqrt{9^2 - 0} = 18$. Consistent.

Final answer: $18$ cm.

Why Chords Show Up Far Beyond Geometry Class

The chord is one of those quiet ideas that turns out to be load-bearing across engineering and design.

• Arch and bridge construction. The straight span across the base of a circular arch is a chord; its length sets how wide the arch can reach, and the chord-to-rise ratio determines how much load the arch carries.

• Aircraft and turbine wings. In aerodynamics the chord of a wing is the straight line from its leading edge to its trailing edge — the same idea, borrowed: a straight segment cutting across a curved profile. Wing performance is quoted per unit chord length.

• Gears and cams. Machinists measure the chordal width of a gear tooth — a straight caliper distance across a curved tooth profile — because you cannot lay a ruler along a curve.

• Surveying circular plots. A straight fence line across a curved boundary is a chord, and the $2\sqrt{r^2 - d^2}$ formula tells the surveyor its length without walking the arc.

The chord properties were proved in Euclid's Elements Book III (c. 300 BCE), and the chord function — the length of a chord for a given angle — was the very first trigonometric table, compiled by Hipparchus (c. 190–120 BCE, Greece) centuries before the sine was named.

Tripping Points to Avoid

Mistake 1: Confusing a chord with the diameter

Where it slips in: Reading a problem that says "the longest chord."

Don't do this: Treat every chord as if it passes through the centre.

The correct way: A chord becomes a diameter only when it passes through the centre. The diameter is the longest chord, but most chords are shorter and sit off to one side.

Mistake 2: Using the full chord instead of the half-chord

Where it slips in: Any problem using the perpendicular-distance right triangle.

Don't do this: Put the whole chord length into $\sqrt{r^2 - (\text{chord})^2}$.

The correct way: The perpendicular from the centre bisects the chord, so the right triangle uses half the chord. The full chord gives a negative inside the root — an impossible value that flags the error.

Mistake 3: Forgetting a chord must have both endpoints on the curve

Where it slips in: Identifying chords in a figure full of segments.

Don't do this: Call a radius or a tangent segment a chord.

The correct way: A chord has both endpoints on the circle. A radius has one endpoint at the centre; a tangent touches at only one point. Neither is a chord.

The Short Version

• A chord is a straight line segment joining two points on a curve, most often a circle's circumference.

• The diameter is a chord — the longest one — and it is the only chord that passes through the centre.

• Chord length is $2\sqrt{r^2 - d^2}$ from the perpendicular distance, or $2r\sin(\theta/2)$ from the central angle.

• A perpendicular from the centre bisects the chord, so chord-length problems use the half-chord in the right triangle.

• Chords reach far past geometry class — into arches, wings, gears, and surveying — wherever a straight line crosses a curve.

Practice These Before Moving On

1. A circle has radius $17$ cm; a chord is $8$ cm from the centre. Find the chord's length.

2. A chord of length $16$ cm sits in a circle of radius $10$ cm. How far is it from the centre?

3. A chord subtends a central angle of $90°$ in a circle of radius $6$ cm. Find its length.

If problem 2 gave you a square root of a negative number, return to Mistake 2 and use the half-chord.

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