KEVOS® MAKEOS · Geometry
Spherical Triangle and Great-Circle Distance
What are the angles and area of this spherical triangle, or the great-circle distance between two points?
Oblique spherical triangle solved from its three sides (as arc angles): the three spherical angles by the law of cosines, spherical excess and surface area; or the great-circle distance and initial bearing between two latitude and longitude points on a sphere.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Calculation | choice | Solve a triangle, or measure between two points. |
| Sphere radius | mm | Radius of the sphere (6,371 km for the earth, in the same length unit you want back). |
| Side a (arc angle) | ° | Side a as the angle it subtends at the centre. |
| Side b (arc angle) | ° | Side b as an angle. |
| Side c (arc angle) | ° | Side c as an angle. |
| Latitude 1 | ° | Latitude of point 1 (south negative). |
| Longitude 1 | ° | Longitude of point 1 (west negative). |
| Latitude 2 | ° | Latitude of point 2. |
| Longitude 2 | ° | Longitude of point 2. |
Method and formulas
- Law of cosines: cos A = (cos a − cos b cos c) ÷ (sin b sin c), and likewise for B and C.
- Area = R² × (A + B + C − 180°) in radians — the handbook’s Area = R² π ÷ 180 × spherical excess.
- Great circle: central angle by the haversine formula; distance = R × angle; initial bearing from the spherical sine and cosine rules.
- Example: sides of 90°, 90° and 120° on a 10 mm sphere enclose 209.44 mm² (one sixth of the sphere, as in the handbook).
Limitations
- Perfect sphere; the earth is slightly flattened (distances within about 0.5%).
- Sides must satisfy the spherical triangle inequalities and sum to less than 360°.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Lay out panels on a dome or spherical tank
- Find the distance between two sites on the earth
- Check spherical geometry for optics or radomes
Who it is for
- Engineers
- Technicians
- Engineering students
What you get
- Spherical angles
- Spherical excess and area
- Great-circle distance and bearing
Interpreting the result
- Arc length of each side = R × side angle in radians.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 115–118. Topic reference; no source tables or recommended settings are reproduced.
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