Chords of a Circle

Joining Two Points on a Circle with a Segment

A chord is a line segment whose two endpoints lie on a circle. A diameter is the special chord that passes through the center. Other chords do not need to pass through it.

Chords $AB$ and $CD$ subtend equal central angles, so they have equal length. Chord $EF$ subtends a larger central angle and is longer.

How Chord Length and Centre Distance Relate

A chord's length and its distance from the centre move together. A smaller distance produces a longer chord, and the two quantities convert into each other through the Pythagorean theorem.

Equal Length Chords

Two chords of equal length have the same distance from the center of the circle. Conversely, two chords at the same distance from the center have equal length.

If $PQ = RS$, then the distance from center $O$ to chord $PQ$ equals the distance from $O$ to chord $RS$, that is $d_1 = d_2$.

Line from Center Perpendicular to Chord

A line drawn from the center of a circle perpendicular to a chord divides the chord into two equal parts.

In the diagram above, $OM \perp AB$ and $M$ is the midpoint of chord $AB$, so $AM = MB$.

Chord Length

A chord and its central angle form an isosceles triangle with two radii. The formula below comes from splitting that triangle in half. The half-triangle gives half the chord length as $r\sin(\theta/2)$.

Chord Length Formula

Here:

• $r$ = radius of the circle
• $\theta$ = the smaller central angle subtending the chord. Use the calculator mode that matches the angle unit

As the smaller central angle $\theta$ increases from $0$ to $180^\circ$, the factor $\sin(\theta/2)$ increases from $0$ to $1$. The chord length therefore increases from $0$ to the diameter $2r$.

Distance of Chord from Center

The distance of a chord from the center of the circle can be calculated using the formula:

Or if the chord length $l$ is known:

Intersecting Chords Theorem

If two chords intersect inside a circle, then the product of the segments of one chord equals the product of the segments of the other chord.

In the diagram above, the following holds: $PA \times PB = PC \times PD$

Inscribed Angles in the Same Segment

Inscribed angles that subtend the same chord have equal measures when their vertices lie on the same side of that chord.

In the diagram above, $\angle ACB = \angle ADB$ because both vertices lie on the same side of chord $AB$ and both angles subtend that chord.

Apothem

The apothem of a chord is the perpendicular segment from the center of the circle to the chord. Its length is the shortest distance between the center and the chord.

The length of the apothem can be calculated using the formula:

Here:

• $a$ = length of apothem
• $r$ = radius of the circle
• $l$ = length of the chord
• $\theta$ = central angle

Parallel Chords

Two parallel chords in a circle have special properties.

If $AB \parallel CD$, then arc $AC$ equals arc $BD$.

Practice Problems

Each problem gives one chord measurement and asks for another. Identify which relation applies, then compute the missing value.

1. A circle has a radius of $10 \text{ cm}$. If the central angle subtending a chord is $60^\circ$, determine:

   • The length of the chord
   • The distance of the chord from the center of the circle

2. Two chords $AB$ and $CD$ intersect at point $P$ inside a circle. If $PA = 4 \text{ cm}$, $PB = 6 \text{ cm}$, and $PC = 3 \text{ cm}$, find the length of $PD$.

3. In a circle with radius $13 \text{ cm}$, there is a chord of length $24 \text{ cm}$. Calculate the distance of this chord from the center of the circle.

4. Two parallel chords in a circle are $3 \text{ cm}$ and $4 \text{ cm}$ away from the center respectively. If the radius of the circle is $5 \text{ cm}$, determine the lengths of both chords.

5. Prove that the longest chord in a circle is the diameter.

Worked Solutions

1. Calculating chord length and its distance from center

   Given: $r = 10 \text{ cm}$, $\theta = 60^\circ = \frac{\pi}{3}$ rad

   $$\begin{aligned} l &= 2r \sin\left(\frac{\theta}{2}\right) \\ &= 2 \times 10 \times \sin\left(\frac{\pi/3}{2}\right) \end{aligned}$$

   $$\begin{aligned} l &= 20 \times \sin(30^\circ) \\ &= 20 \times 0.5 \\ &= 10 \text{ cm} \end{aligned}$$

   $$d = r \cos\left(\frac{\theta}{2}\right) = 10 \times \cos(30^\circ)$$

   $$d = 10 \times \frac{\sqrt{3}}{2} = 5\sqrt{3} \text{ cm}$$

2. Intersecting chords theorem

   Given: $PA = 4 \text{ cm}$, $PB = 6 \text{ cm}$, $PC = 3 \text{ cm}$

   $$PA \times PB = PC \times PD$$

   $$4 \times 6 = 3 \times PD$$

   $$24 = 3 \times PD$$

   $$PD = 8 \text{ cm}$$

3. Calculating distance of chord from center

   Given: $r = 13 \text{ cm}$, $l = 24 \text{ cm}$

   $$d = \sqrt{r^2 - \left(\frac{l}{2}\right)^2}$$

   $$d = \sqrt{13^2 - 12^2}$$

   $$d = \sqrt{169 - 144}$$

   $$d = \sqrt{25} = 5 \text{ cm}$$

4. Parallel chords

   Given: $r = 5 \text{ cm}$, $d_1 = 3 \text{ cm}$, $d_2 = 4 \text{ cm}$

   For the first chord:

   $$\begin{aligned} l_1 &= 2\sqrt{r^2 - d_1^2} \\ &= 2\sqrt{25 - 9} \\ &= 2\sqrt{16} \\ &= 8 \text{ cm} \end{aligned}$$

   For the second chord:

   $$\begin{aligned} l_2 &= 2\sqrt{r^2 - d_2^2} \\ &= 2\sqrt{25 - 16} \\ &= 2\sqrt{9} \\ &= 6 \text{ cm} \end{aligned}$$

5. Proof that diameter is the longest chord

   For any chord, take $\theta$ to be the smaller central angle, so $0^\circ\leq\theta\leq180^\circ$:

   $$l = 2r \sin\left(\frac{\theta}{2}\right)$$

   The maximum value of $\sin\left(\frac{\theta}{2}\right) = 1$ is achieved when $\frac{\theta}{2} = 90^\circ$, that is $\theta = 180^\circ$.

   When $\theta = 180^\circ$, the chord passes through the center of the circle (diameter) with length:

   $$l_{\max} = 2r \times 1 = 2r$$

   Therefore, the diameter is the longest chord.