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Chord Length Formula with Problem Solution & Solved Example


The circle is taken as an integral part of geometry and the chord length is defined as the line segment whose endpoints lie on the circumference of a circle. Practically, a circle could have infinite chords. The chord is always seen within a circle and the diameter is the longest chord inside a circle. The chord of a circle is a straight line that connects any two points on the circumference of a circle.

The chord length formula in mathematics could be written as given below. The chord length formulas vary depends on what information do you have about the circle. If you know the radius or sine values then you can use the first formula. The second formula is a variation of the Pythagorean theorem and it can be used for calculating the length of a chord as well.

Chord Length Formula

\[\LARGE Chord\;Length=2r\sin \left (\frac{c}{2}\right )\]
\[\LARGE Chord\;Length=2\sqrt{r^{2}-d^{2}}\]
Where,r is the radius of the circle
c is the angle subtended at the center by the chord
d is the perpendicular distance from the chord to the circle center
sin is the sine function

Here, r is the radius of a circle, c is angle subtended at the center by the chord, d is the perpendicular from chord to the center of a circle, and sin is the sine trigonometry function. If any line that does not stop at the circumference of a circle instead it is extended to infinity then it is called as the secant.

Practically, this is not possible finding the chord length if you cannot measure the angle. Still, you have the flexibility of using trigonometry functions here but they are little bit difficult to understand. At the same time, it is easy to calculate the chord length if you know the radius of the circle and one of the variables.

Question 1:Find the chord of a circle where radius is 7 cm and perpendicular distance from chord to center is 4 cm?

Solution:

Given radius, r = 7 cm

and distance, d = 4 cm

\[\ Chord\;Length=2\sqrt{r^{2}-d^{2}}\]
\[\ Chord\;Length=2\sqrt{7^{2}-4^{2}}\]
\[\ Chord\;Length=2\sqrt{49-16}\]
\[\ Chord\;Length=2\sqrt{33}\]
\[\ Chord\;Length=2\times 5.744 \]

Chord length = 11.48 cm


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