Table of Contents
Last modified on August 3rd, 2023
The chord of a circle is a straight line joining two points on the circumference of the circle. The diameter that passes through the center of the circle is the longest chord of the circle.
Shown below are 3 chords AB, CD, and EF. AB is also the diameter, so it is the longest chord of the given circle.
There are two basic formulas to find the length of the chord of a circle. Both the formulas give the same result but are used based on the information provided. Let us discuss each of them in detail with solved examples.
1) When the Radius and the Perpendicular Distance from the Center is Known
If the length of the radius and the distance between the center and the chord is given, then the formula to find the chord is given below.
Let us solve an example to understand the concept better.
Find the length of the chord if the radius of a circle is 16 cm, and the perpendicular distance from the chord to the center is 8 cm.
As we know,
Length (L) of chord = 2√(r2 – d2), here r = 16 cm, d = 8 cm
= 2√(162 – 82) cm
= 2√(256 – 64) cm
= 2√192 cm = 27.71 cm
The perpendicular distance from the center of a circle to the chord is 12 inches. Calculate the chord’s length if the circle’s diameter is 26 inches.
As we know,
Length (L) of chord = 2√(r2 – d2), here r = 26/2 = 13 inches, d = 12
= 2√(132 – 122) inches
= 2√(169 – 144) inches
= 2√25 inches = 10 inches
2) When the Radius and the Central Angle is Known
If the radius and the central angle of the chord are known, then the formula to find the chord is given below.
Calculate the length of the chord AB in the given diagram.
As we know,
Length (L) of the chord = 2 × rsin (C/2), here r = 26 m, C = 75°
= 2 × 26 × sin (75°/2)
= 52 × sin 37.5°
= (52 × 0.60) m
= 31.2 m
Find the length of the cord and the central angle of the chord in the given diagram.
As we know,
Length (L) of chord = 2√(r2 – d2), here r = 50 m, d = 20 m
= 2√(502 – 202) m
= 2√(2500 – 400) m
= 2√2100 cm = 91.65 m
Now, for finding the central angle,
Length (L) of the chord = 2 × rsin (C/2), here r = 26 m, C = 75°
=> 91.65 = 2 × 50 × sin (C/2)
=> 91.65 =100 sin (C/2)
=> 0.9165 = sin (C/2)
=> sin-1 0.9165 = C/2
=> 66.41 = C/2
=> C = 132.82°
The relationship between the different chords and the angle subtended by them to the center of the circle can be established using the theorems given below.
Ans. Yes, every diameter of a circle is a chord because its two endpoints lie on the circumference of the circle.
Ans. A circle can have an infinite number of equal chords.
Last modified on August 3rd, 2023
Clearly explained and interesting lesson