Calculate the chord length of a circle from central angle, arc length, or sagitta. Essential for geometry, engineering, and circular design.
Last updated: April 2026 | By Patchworkr Team
A chord is a straight line segment connecting two points on a circle's circumference. The longest possible chord is the diameter, which passes through the circle's center.
Related terms:
Identify your starting information: do you know the central angle, arc length, or sagitta? Why: Each method leads to a different formula. In real applications, you might measure the arch height (sagitta) at a construction site, or know the angle from a blueprint, or calculate arc length from specifications. The available data determines your calculation path.
Determine the radius r of the circle. This is essential for all three methods. Why: The radius scales everything—a chord in a small circle is short; the same central angle in a large circle produces a much longer chord. The radius is the fundamental reference for all circular geometry.
Why: Each formula is derived from the circle's geometry. The sine method uses right-triangle trigonometry. The sagitta method uses the Pythagorean theorem applied to the radius, sagitta, and half-chord forming a right triangle.
Plug your values into the appropriate formula and compute. Ensure angles are in radians (convert degrees if needed: radians = degrees × π/180). Why: Most math formulas require radians, not degrees. Mixing units causes 57-fold errors. Always verify your angle units before plugging into trigonometric functions.
Check that the chord length is reasonable: it must be ≤ diameter (2r), and for small angles, the approximation c ≈ r·θ should be close. Cross-check using the other formulas (calculate arc length from angle, then verify chord matches). Why: Calculation errors are easy to make. Sanity checks catch them immediately. For structural engineering or manufacturing, a 5% error might waste thousands of dollars—verification is critical.
Bridge Arch Design
c = 2r·sin(θ/2), where r is radius and θ is the central angle in radians. For a 90° angle, chord = r√2.
Sagitta (or height) is the perpendicular distance from the chord's midpoint to the arc. It's used in archery, architecture, and optics.
A chord is the straight line connecting two points. An arc is the curved path along the circle between the same points. The arc is always longer unless the chord is a diameter.
The diameter is the longest chord, passing through the center. Its length is 2r, and it subtends a 180° central angle.
Yes! If you know the sagitta (h) and radius (r), use c = 2√(2rh - h²). Or if you know the arc length, first calculate the angle.
The formula uses half the central angle because we're finding the hypotenuse of a right triangle formed by half the chord and the radius.
Rearrange the formula: θ = 2·arcsin(c/(2r)). This gives the angle in radians.
Architecture (arches, domes), civil engineering (roads, bridges), astronomy (planetary orbits), manufacturing (circular parts), and surveying.
Related Tools