Earth arc length from central angle calculator

Enter the angle subtended at the sphere's centre and choose the radius in metres. The calculator converts degrees to radians, applies the exact spherical relation s = r × theta, and returns the arc in metres and kilometres. It works with the mean Earth radius, another accepted Earth-radius convention, or a custom sphere used for education and modelling.

Choose the central angle and spherical radius

A central angle is measured at the centre of a sphere between two radial lines that reach the endpoints of an arc. Enter that value as central_angle_deg. A 0-degree angle represents no surface separation, 90 degrees represents one quarter of a complete circumference, 180 degrees reaches the antipodal point, and 360 degrees traces a full circumference back to the starting direction. Then enter radius_m, the radius of the spherical model in metres. For a general Earth-distance estimate, a commonly used mean radius is 6,371,008.8 metres, but this capability deliberately requires the radius so the chosen convention remains explicit. You may instead provide an equatorial, polar, authalic, or project-specific radius, provided it is positive and within the documented limit. Keeping the radius in the input prevents hidden assumptions and makes results reproducible. The calculation describes a perfect sphere, so both the angle and radius must refer to the same geometric model. It does not infer a radius from coordinates, an ellipsoid, elevation, or a named planet.

How the great-circle arc length is calculated

The governing relation is s = r × theta, where s is surface arc length, r is sphere radius, and theta is the central angle expressed in radians. Because the input angle is in degrees, the calculator first multiplies central_angle_deg by pi divided by 180. It then multiplies that radian value by radius_m. No numerical integration, route lookup, map projection, or network request is needed. The response preserves the supplied central_angle_deg and radius_m, reports central_angle_rad for inspection, and returns arc_length_m together with arc_length_km. Values are rounded deterministically to nine decimal places so repeated browser and API calculations have a stable representation while retaining ample precision for the spherical model. The formula is linear: doubling the angle doubles the distance, and doubling the radius does the same. This also provides useful checks. At 180 degrees the result should equal pi times the radius, while at 360 degrees it should equal two times pi times the radius, the sphere's complete circumference.

Interpret the result as a spherical surface distance

The returned arc length is the distance along a great circle associated with the supplied central angle. For angles from 0 through 180 degrees, it can represent the shorter or equal great-circle separation between two points on a sphere. Angles above 180 degrees describe the longer complementary arc and remain valid when that is the intended path. This distinction is important because coordinate-based distance calculators usually choose the shortest geodesic automatically, whereas this calculator respects the angle you provide. The result is not an ellipsoidal geodesic and does not account for Earth's flattening, terrain height, roads, currents, restricted routes, or measurement uncertainty. Different legitimate Earth radii therefore produce slightly different distances for the same angle. Use a radius that matches your source or standard, and record it beside the output when results will be compared later. For surveying or high-precision navigation, use an ellipsoidal method with coordinates and a stated datum. For lessons, quick global estimates, spherical simulations, and conversions from an already-known angular separation, this direct calculation is transparent and appropriate.