The document provides a comprehensive overview of the chord of a circle, including its definition, properties, and formulas for calculating its length. It explains key terms related to circles, describes important properties of chords, and presents methods for finding the length of chords using trigonometry and perpendicular distances. Additionally, the document covers theorems pertaining to chords, emphasizing relationships between chord lengths and angles subtended at the center of a circle.

1. Chord of a Circle: Definition, Formula, Theorem &
Examples
Learn everything about the chord of a circle along with its terms like
chord length formula, chord length, examples using chord length formula,
and a lot more. We have designed this blog format for you in an easy -to-
understand step-by-step method considering your queries and
complexities. Reading this would make you feel like you are reading notes
customized by your private math tutor. So, let’s start learning!
What is the Chord of a Circle?
A line segment that connects or joins two points on a circle’s
circumference is known as the chord of a circle. By definition, the
diameter will be the longest chord of the circle as it passes through the
mid or center of the circle, cutting it into two equal halves. Chord’s
endpoints will always lie on the circumference of a circle.

2. Note: circumference is the distance surrounding the circle.
Following is the figure of a circle with the longest chord (diameter) to h elp
you understand the chord of the circle more clearly.
Here, AD represents the circle’s diameter (the longest chord) with ‘O’ as
the center. Also, ‘OG’ represents the radius of the circle and ‘BP’
represents a chord of the circle.
Key Parts of a Circle
Before delving into the details, formulas, and questions regarding the
chord of the circle, here are the main key parts of a circle. Y ou just need
to know:
• Radius: the distance between the circumference of the circle or the
center of the circle is called the radius of the circle.

3. • Diameter: twice the length of the radius of a circle is called the
diameter of a circle. It is a line that me ets the ends of the
circumference of the circle and passes through the center.
• Arc: in the circle, the part of the circumference is known as an arc.
A circle has two arcs: major arc and minor arc.
• Major Segment: when a circle is enclosed by a chord and the major
arc, its largest part is known as the major segment.
• Minor Segment: when a circle is cut by a chord and the minor arc,
its smaller part is known as the minor segment.
Properties of the Chord of a Circle
Following are a few important properties of the chord of a circle, you
should know:
1. Only one circle passes through the three collinear points.
2. A chord becomes a secant when it is extended infinitely on both
sides.
3. The perpendicular to a chord bisects the c hord when drawn from
the center of the circle.
4. Chords which are equidistant from the center of the circle have
equal length. Or you can say equal chords are equidistant from the
center of a circle.
5. The chord of a circle divides the circle into two regions, also known
as the segments of the circle: the minor segment and the major
segment.
6. Equal chords of a circle have equal angles.

4. 7. If two chords of a circle intersect inside it, then the product of the
segment’s lengths of one chord is equal to the lengths of the
segments of the other chord.
What is the Chord Length Formula?
Calculating the length of the chord of any circle is important to solve
some questions and theorems related to circles. How to find the chord
length of a circle differs as there are two ba sic formulas for it.
• For calculating the chord length using perpendicular distance from
the center, apply:
Chord Length = Clen = 2 * √ (r2 − d2)
• For calculating the chord length using trigonometry, apply :
Chord Length = Clen = 2 * r * sin (θ/2)

5. Here,
r = the radius of a circle
θ = the angle subtended at the center by the chor d
d = the perpendicular distance from the chord to the center of a circle
Tip: How to find the right formula to calculate the chord length of a circle?
• If you the radius and the perpendicular distance from the chord to
the circle center is given then the formula would be 2 * √ (r2 − d2).
• If you know the value of angle subtended at the center by the chord
and the radius of the circle then the formula to find the chord
length would be 2 * r * sin (c/2).
Chord Length Formula Example Questions
Question 1: Calculate the radius of the circle if 2 is the perpendicular
distance between the chord and the center and 5 is the length of the
chord. Use the chord of length formula.

6. Solution:
Given that,
Length of the chord= Cl e n = 5
The perpendicular distance between the chord and the center= d = 2
Radius of the circle = r =?
We will find the radius using;
⇒ Cl e n = 2 * √ (r2
− d2
)
(Put given values)
⇒ 5 = 2 * √ (r2
− 22
)
⇒ 2.5 = √ (r2
− 4)
Now, sqrt on both sides
⇒ 6.25 = r2
− 4
⇒ 6.25 + 4 = r2
⇒ 10.25 = r2
⇒ √ (10.25) = r
⇒ 3.2 = r
Hence, the radius of the circle is about 3.2
Question 2: Jack is eating dinner with his family at a restaurant nearby.
He ordered spaghetti for himself, and it was served on a plate of radius 7
inches. What would be the angle swept out the chord if the spaghetti
strand is 5 inches long? Apply chord length formul a.
Solution:
Formula
Cl e n = 2 * r * sin (θ/2)

7. where,
Cl e n = length of chord = 5
r = radius = 7
(Put values in the formula)
⇒ 5 = 2 x 7 x sin (θ/2)
⇒ 5 = 14 x sin (θ/2)
⇒ 5 / 14 = sin (θ/2)
⇒ sin− 1
(5/14) = θ/2
⇒ 0.365 = θ/2
⇒ θ = 0.73
Hence, the angle of spaghetti is 0.73 radians.
Theorems: Chord of a Circle
Theorem 1: Chords with equal lengths subtend equal angles at the center
of a circle. Prove equal chords, equal angles of a circle.

8. Proof: ∆BOC and ∆XOY
Given that,
⇒ BC = XY
chords of equal length
⇒ OB = OC = OX = OY
radius of the same circle
⇒ ∆BOC ≅ ∆XOY
side-side-side axiom of the congruence
⇒ ∠ BOC = ∠ XOY
the congruent parts of congruent triangles, CPCT
Hence, proved.

9. Theorem 2: The measure of angles subtended by the chord at the center of
a circle equals the length of the chords. Prove equal angles equal chords.
Proof: ∆BOC and ∆XOY
⇒ ∠ BOC = ∠ XOY
given that, equal angles subtend at center O of the circle
⇒ OB = OC = OX = OY
radius of the same circle
⇒ ∆BOC ≅ ∆XOY
SSS or side-side-side axiom of the congruence
⇒ BC = XY
The congruent parts of congruent triangles, CPCT

10. Hence, proved.
Theorem 3: The perpendicular to a chord, drawn from the center of the
circle, bisects the chord. Prove equal chords equidistant from the center of
the circle.
Proof:
Given that, chords AC and BD are equal in length
Now, join A and B with center O and drop perpendiculars from O to the
chords AC and BD
⇒ AM = AC/2 and BN = BD/2
These are the perpendicular from the center, bisect the chord
In △OAM and △OBN
⇒ ∠1 = ∠2 = 90°
OM ⊥ AC and ON ⊥ BD

11. ⇒ OA = OB
Radius of the same circle
⇒ OM = ON
Given that
⇒ △OPB ≅ △OND
The RHS Axiom of Congruency
⇒ AM = BN
Two corresponding parts of congruent triangle
⇒ AC = BD
Hence, proved.
Note: We have mentioned the statements and reasons respe ctively in the
theorems.