Circle Chord & Height
Compute chord c and sagitta (height) h of a circle arc given radius r and angle θ (or arc length s). c = 2r·sin(θ/2), h = r − r·cos(θ/2).
—
Chord of a circle from sagitta
Take a circle of radius r and the height h of a circular segment. That height is also known as the sagitta: the perpendicular distance from the midpoint of the chord up to the arc. The chord length works out to c = 2·√(r² - (r - h)²) = 2·√(h·(2r - h)). Plug in r = 5 m and h = 2 m and you get c = 2·√(2·(10 - 2)) = 2·√16 = 8 m. Where does it come from? Just the Pythagorean theorem applied to the right triangle whose hypotenuse is the radius and whose legs are half the chord and (r - h). At h = r the chord becomes a diameter, so c = 2r, and as h approaches 0 the chord shrinks to nothing.
Applications
Architecture leans on sagitta-chord geometry to lay out arches, vaults and domes. The mason knows the span (the chord) and the rise (the sagitta), and has to work back to the radius before cutting a template. Construction of curved formwork and pipe sections runs into the same need. And celestial navigation historically reduced solar altitude observations with related circular-segment relations.
FAQ
How do I get the radius from chord and sagitta? Run the formula backwards: r = (c² + 4h²)/(8h).
What if h > r? The segment is then bigger than a semicircle and what you entered is the height of the "major" segment. The same formula holds, since c = 2·√(h·(2r - h)) ≥ 0 only requires h ≤ 2r.
Sagitta vs apothem? The sagitta runs from the chord up to the arc, while the apothem runs from the centre down to the chord. Add the two together and you get r.
Related Tools
Hydraulic Jump Sequent Depth
Calculate the sequent (conjugate) depth downstream of a hydraulic jump, y₂ = (y₁/2)·(√(1 + 8·Fr₁²) − 1), from the upstream depth y₁ (supercritical) and the incoming Froude number Fr₁. The hydraulic jump is the abrupt transition from fast, shallow (supercritical) to slow, deep (subcritical) flow, with strong turbulence and energy dissipation. This Bélanger equation, from momentum conservation, is the basis for designing stilling basins downstream of spillways and gates: water descending a spillway arrives at very high (supercritical) velocity and must be decelerated before returning to the river, otherwise it erodes the bed catastrophically. The sequent depth y₂ sets the required basin depth for a stable jump. Enter the upstream depth and the Froude number.
Approach Surface Height
Calculate the height of an approach surface (or other obstacle limitation surface) at a given distance, h = (gradient ÷ 100) · distance, from the ramp gradient (%) and the horizontal distance from the surface origin (m). Obstacle Limitation Surfaces (OLS) are imaginary inclined planes projected from runway thresholds and around runways, defined by ICAO, delimiting the airspace that must stay clear of obstacles for safe landing and takeoff. The approach surface, for example, rises at a typical 2% (1:50) gradient from the runway strip end; any object (building, antenna, tree, terrain) penetrating it is an obstacle to be removed, lowered, marked/lit or, ultimately, leading to operational restrictions. This calculation gives the maximum allowed surface height at each point, to compare with the actual height of existing or proposed obstacles around the airport — the basis of land-use control in airport protection zones and the assessment of new developments. Enter the gradient and the distance.
FDM Layer Height Calculator
Suggests FDM layer height in mm from nozzle size and quality target.
The results provided by this tool are for general informational and educational purposes only and do not constitute professional, financial, medical, legal, tax or accounting advice. Always confirm important decisions with a qualified professional and official sources.