The chord length in a circle can be found using different methods depending on the available information. One common method involves using the perpendicular distance from the center of the circle to the chord and the radius of the circle.
Chord Length Formula Using Perpendicular Distance
If you know the radius of the circle (r) and the perpendicular distance (d) from the center of the circle to the chord, you can use the following formula to find the chord length:
Chord Length = 2 × √(r2 − d2)
Derivation of the Formula
This formula is derived from the Pythagorean theorem. Imagine a circle with a chord and a line drawn from the center of the circle perpendicular to the chord. This line bisects the chord. The radius of the circle (r) forms the hypotenuse of a right-angled triangle, with one side being half the chord length and the other side being the perpendicular distance (d) from the center to the chord.
Therefore:
(r)2 = (d)2 + (Half of Chord Length)2
(Half of Chord Length) = √(r2 − d2)
Chord Length = 2 × √(r2 − d2)
Example
Suppose you have a circle with a radius of 5 cm, and the perpendicular distance from the center to the chord is 3 cm. Then, the chord length can be calculated as follows:
Chord Length = 2 × √(52 − 32)
Chord Length = 2 × √(25 − 9)
Chord Length = 2 × √16
Chord Length = 2 × 4
Chord Length = 8 cm
Summary
In summary, finding the chord length in a circle often involves knowing the radius and the perpendicular distance from the center to the chord, allowing you to apply the formula: 2 × √(r2 − d2).
