Calculate sagitta (sag depth/height), radius of curvature, chord width, and arc geometry for optics, lenses, curved mirrors, and circular arches.
Select geometry profile, calculation target, and dimension unit
Spherical for standard optics/arches • Parabolic for astronomical telescope mirrors
Center-to-surface sphere radius
Linear span or lens diameter
Center depth or rise from chord
Horizontal distance between spring points
Arc Angle: 7.17° (0.1252 rad) • R = 64.00 in
Focal length: 32.00 in
+0.16% over chord
0.1252 radians
Spherical vs Parabolic diff
Amateur telescope makers (ATMs) and optical engineers grinding mirror blanks need to measure center curve depth with spherometers accurate to microns. This tool provides the exact spherical and parabolic sagitta targets needed before polishing begins.
Carpenters and stone masons framing a curved doorway, segmental arch, or barrel ceiling know the doorway span and ceiling rise, but need the swing arm radius for their trammel point compass. Our solver reverses the geometry instantly.
The parabolic approximation $s \approx r^2 / (2R)$ is fine for shallow curves, but at steep f-numbers ($< f/5$), edge deviation between spherical and parabolic curves causes severe wavefront error. Our calculator highlights the exact difference ($\Delta s$).
Metal fabricators rolling angle iron or structural tubing into circular arcs need chord-and-sagitta measurements to verify radius compliance on job sites using standard string lines and digital calipers.
Choose whether to calculate the arc depth (sagitta $s$) from radius and diameter, or reverse-calculate the circle radius ($R$) from arch span and rise.
Select a spherical surface (conic constant $k = 0$) for standard lenses, or a true paraboloid ($k = -1$) for telescope primary mirrors.
Enter dimensions in your preferred unit: inches, millimeters, centimeters, feet, or meters. The tool automatically balances units.
Examine the exact sagitta height, subtended arc angle (degrees/radians), arc perimeter distance, and focal ratio ($f/\#$).
The term sagitta originates from the Latin word for "arrow", depicting the arrow nocked onto a bowstring (the chord) and pointing toward the curved bow (the circular arc). In Euclidean geometry, the perpendicular bisector of any chord passes directly through the circle's center, establishing the fundamental right triangle relationship:
Solving for $s$ produces the exact quadratic root: $s = R - \sqrt{R^2 - r^2}$. In lens manufacturing and ophthalmology, when the aperture ratio is small ($r \ll R$), neglecting the higher-order $s^2$ term yields the standard parabolic formula $s \approx \frac{r^2}{2R}$.
Solves precise circular geometry via the Pythagorean theorem: s = R - sqrt(R² - (C/2)²).
Calculate sagitta height from radius and chord, radius from sagitta and chord, or chord from radius and sagitta.
Ideal for curved brick lintels, round barrel vault framing, bent glass facades, and lens manufacture.
High-precision floating point math supports micro-inch optical engineering and large-scale architectural spans.
Sagitta (commonly abbreviated as 'sag') is the perpendicular distance from the center of a circular chord to the highest point of the arc. In optics, it represents the physical center thickness difference or curve depth of a concave or convex optical lens or mirror relative to its clear aperture diameter.
The exact formula for a spherical surface is: s = R - sqrt(R^2 - r^2), where s is the sagitta, R is the radius of curvature, and r is the semi-aperture or half-chord width (r = c / 2 or D / 2). For shallow curves where R >> r, the parabolic approximation s ≈ r^2 / (2R) is commonly used.
To find radius (R) from chord width (c) and sagitta (s), use the intersecting chord theorem formula: R = (s / 2) + (c^2 / (8s)). For example, if a chord is 10 inches and the sagitta is 1 inch: R = (1 / 2) + (100 / 8) = 0.5 + 12.5 = 13.0 inches.
For a true parabola, the sagitta is exactly s = r^2 / (2R) across the entire aperture. For a sphere, the curvature steepens towards the edge according to s = R - sqrt(R^2 - r^2). At large aperture ratios (fast f-numbers), the spherical sag is slightly deeper than parabolic sag due to spherical aberration.
In Euclidean geometry, the semi-diameter r cannot exceed the radius R of the sphere. If r > R, the term (R^2 - r^2) inside the square root becomes negative, resulting in an imaginary number because a circle of radius R cannot span a chord wider than its full diameter 2R.
Spiral risers, treads, and helical rotation angles.
Circular disc, rectangular plate, and alloy weights.
Segmental concrete blocks, courses, and drainage.
Equal spindle spacing with IRC 4-inch sphere check.
