Calculate The Length Of A Chord

Calculate the Length of a Chord: A Complete Guide with Formulas and Examples

Understanding how to calculate the length of a chord is a fundamental skill in circle geometry with practical applications in engineering, architecture, design, and even everyday problem-solving. A chord is simply a straight line segment whose endpoints both lie on the circumference of a circle. The longest possible chord is the diameter, which passes through the circle’s center. Whether you’re designing a circular structure, analyzing a wheel, or solving a geometry problem, knowing the precise methods to determine chord length is essential. This guide will walk you through the core concepts, formulas, derivations, and step-by-step calculations, ensuring you can confidently tackle any chord length problem.

What Exactly is a Chord?

Before diving into calculations, it’s crucial to have a clear mental model. A chord connects any two points on a circle’s edge. Its length depends entirely on two key measurements: the circle’s radius (r) and the central angle (θ) subtended by the chord—that is, the angle formed at the circle’s center by lines drawn to the chord’s endpoints. Alternatively, the chord length can be found using the radius and the perpendicular distance (d) from the chord to the circle’s center. This perpendicular line from the center to the chord is always bisected by the chord, splitting it into two equal halves. This property is the geometric key that unlocks all chord length formulas.

The Primary Formula: Using Radius and Central Angle

The most common and direct formula for chord length (c) arises from an isosceles triangle formed by the two radii and the chord.

The Formula: [ c = 2r \cdot \sin\left(\frac{\theta}{2}\right) ] Where:

• c = chord length
• r = radius of the circle
• θ = central angle in radians (If your angle is in degrees, you must convert it to radians first: (\theta_{rad} = \theta_{deg} \times \frac{\pi}{180}))

Step-by-Step Calculation Example

Problem: A circle has a radius of 10 cm. Find the length of a chord that subtends a central angle of 60°.

1. Convert the angle to radians:
   (\theta = 60° \times \frac{\pi}{180} = \frac{\pi}{3}) radians.
2. Apply the formula:
   (c = 2 \times 10 \times \sin\left(\frac{\pi/3}{2}\right) = 20 \times \sin\left(\frac{\pi}{6}\right))
3. Calculate the sine:
   (\sin(\frac{\pi}{6}) = 0.5)
4. Final calculation:
   (c = 20 \times 0.5 = 10 \text{ cm}).

Answer: The chord length is 10 cm. Notice that for a 60° angle in a circle of radius 10, the chord length equals the radius, forming an equilateral triangle.

The Alternative Formula: Using Radius and Perpendicular Distance

Often, you might know how far the chord is from the center (the sagitta or "versine" when measured from the chord to the circumference, but here we use the center-to-chord distance d). This method uses the Pythagorean theorem on the right triangle formed by the radius, half the chord, and the distance d.

The Formula: [ c = 2 \sqrt{r^2 - d^2} ] Where:

• c = chord length
• r = radius
• d = perpendicular distance from the circle’s center to the chord (must be less than r).

Step-by-Step Calculation Example

Problem: A circular table has a radius of 1.5 meters. A chord (the edge of a placemat) is located 1.2 meters from the center of the table. What is the length of this chord?

1. Identify values:
   (r = 1.5 \text{ m}), (d

2. Complete the calculation:
   (d = 1.2 \text{ m})
   (c = 2 \times \sqrt{1.5^2 - 1.2^2} = 2 \times \sqrt{2.25 - 1.44} = 2 \times \sqrt{0.81} = 2 \times 0.9 = 1.8 \text{ m}).
   Answer: The chord length is 1.8 meters.

The Geometric Bridge: Connecting Both Formulas

The two primary