Chord Length Calculator

A chord is a straight line segment whose endpoints both lie on a circular arc. The geometry of a chord is governed by relationships between the circle's radius (r), the central angle (θ) subtended by the chord, and the perpendicular distance (d) from the circle's center to the chord. When you know the radius and the central angle in radians, the chord length (c) is calculated using the formula: c = 2 * r * sin(θ / 2). Alternatively, if you know the radius and the perpendicular distance from the center to the chord, you can apply the Pythagorean theorem via the equation: c = 2 * sqrt(r^2 - d^2). These foundational trigonometric relationships allow you to solve for unknown circle dimensions seamlessly.

Let us walk through a practical calculation to see how the math operates in practice. Imagine you have a circular architectural window with a radius of 5 feet, and you want to install a decorative horizontal brace spanning a central angle of 30 degrees. First, convert the angle from degrees to radians by multiplying by π/180, which yields approximately 0.5236 radians (or simply use the angle directly if your formula applies degrees). Plugging our values into the formula c = 2 * r * sin(θ / 2), we get c = 2 * 5 * sin(30° / 2). Since sin(15°) is approximately 0.2588, the calculation becomes 10 * 0.2588, resulting in a chord length of roughly 2.59 feet. This tells you the exact length of the material needed for your brace.

When working with circle calculations, always double-check whether your trigonometric functions expect angles in degrees or radians to avoid major calculation errors. Remember that the longest possible chord in any circle is the diameter, which passes directly through the center point where the perpendicular distance equals zero. Finally, sketching a quick visual diagram of your circle, radius, and central angle before inputting numbers can greatly help you choose the correct formula variant for your specific problem.