The formula for calculating the length of a chord depends on the information available, primarily involving the circle's radius, the central angle subtended by the chord, or the perpendicular distance from the circle's center to the chord.
Understanding the Chord of a Circle
A chord is a straight line segment whose endpoints both lie on the circumference of a circle. The longest chord in any circle is its diameter.
There are two primary formulas to calculate the length of a chord, based on different known parameters:
Formulas for Calculating Chord Length
Here are the key formulas used to determine the length of a chord:
| Method of Calculation | Formula | Variables Explained |
|---|---|---|
| Using Perpendicular Distance from the Center | Chord Length = 2 × √(r² − d²) |
r = Radius of the circle d = Perpendicular distance from the center to the chord |
| Using Trigonometry (with Central Angle) | Chord Length = 2 × r × sin(c/2) |
r = Radius of the circle c = Central angle subtended by the chord (in radians or degrees) |
In-Depth Explanation of Each Formula
Let's delve deeper into each method to understand its application.
1. Chord Length Using Perpendicular Distance from the Center
This method is derived from the Pythagorean theorem. When a perpendicular line is drawn from the center of a circle to a chord, it bisects the chord. This creates two right-angled triangles where:
- The hypotenuse is the radius (
r). - One leg is the perpendicular distance from the center to the chord (
d). - The other leg is half the length of the chord (
L/2).
From the Pythagorean theorem: r² = d² + (L/2)²
Rearranging to solve for L:
(L/2)² = r² − d²L/2 = √(r² − d²)L = 2 × √(r² − d²)
Practical Insight: This formula is particularly useful when you know the circle's radius and how far the chord is from the center.
- Example: A circle has a radius of 10 cm. A chord is located 6 cm away from the center.
r = 10cmd = 6cmChord Length = 2 × √(10² − 6²) = 2 × √(100 − 36) = 2 × √64 = 2 × 8 = 16cm.
2. Chord Length Using Trigonometry (with Central Angle)
This formula utilizes the central angle subtended by the chord. The central angle (c) is the angle formed by two radii connecting the center to the endpoints of the chord.
When you draw the radii to the endpoints of the chord and then bisect the central angle with a line from the center to the midpoint of the chord, you form two right-angled triangles.
- The hypotenuse of these triangles is the radius (
r). - The angle at the center for each right-angled triangle is
c/2. - The side opposite to this angle is half the chord length (
L/2).
Using the sine function: sin(c/2) = (Opposite Side) / (Hypotenuse) = (L/2) / r
Rearranging to solve for L:
L/2 = r × sin(c/2)L = 2 × r × sin(c/2)
Important Note: Ensure the central angle c is in the correct unit (radians or degrees) as expected by the sin function in your calculator or computational tool.
Practical Insight: This method is ideal when the angle that the chord forms at the center of the circle is known, along with the circle's radius.
- Example: A circle has a radius of 5 cm, and a chord subtends a central angle of 60 degrees.
r = 5cmc = 60degreesChord Length = 2 × 5 × sin(60/2) = 10 × sin(30°) = 10 × 0.5 = 5cm.
For more detailed information on chords and related theorems, you can refer to resources like Byju's: Chord Of A Circle, Its Length and Theorems.
