BIZOMAT 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.
Open the calculatorRuns in your browser. No account needed.
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. |
What each input means
Open an input to see what it is, its units, the worked example and the common mistakes, with a small diagram.
What this means: Calculation
- What it is
- What to solve: a spherical triangle whose three sides you know as arc angles, or the great-circle distance and starting bearing between two latitude and longitude points.
- Units and format
- Choose one of 2 options from the list: Triangle from three sides; Great circle between two points.
- Example
- The form opens with “Triangle from three sides” selected as a worked example. Choose the option that matches your case.
- Get it right
- Triangle mode uses sides a, b and c; great-circle mode uses the two latitudes and longitudes. The others are ignored.
What this means: Sphere radius
- What it is
- The radius of the sphere the triangle or the two points lie on. It turns the angles into a surface area or a distance, in the same length unit you type.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm or more. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 10 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Distance is proportional to the radius and triangle area to its square; the angles do not change.
- Get it right
- Results are reported in the length unit of this radius: type kilometres for a distance in kilometres, whatever the unit label shows.
What this means: Side a (arc angle)
- What it is
- Side a of the spherical triangle, given as the angle it subtends at the centre of the sphere rather than as a length along the surface.
- Units and format
- Enter a number in °: degrees of angle. The form accepts 0.000001° to 179.999°. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 90° as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- To convert an arc length along the surface to this angle, divide it by the radius and convert radians to degrees.
What this means: Side b (arc angle)
- What it is
- Side b of the spherical triangle, the arc between the other two corners, given as the angle it subtends at the sphere’s centre.
- Units and format
- Enter a number in °: degrees of angle. The form accepts 0.000001° to 179.999°. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 90° as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- The sum of any two sides must exceed the third, and all three together must be less than 360°.
What this means: Side c (arc angle)
- What it is
- Side c of the spherical triangle, given as its central angle. The angle C reported by the tool is the corner opposite this side.
- Units and format
- Enter a number in °: degrees of angle. The form accepts 0.000001° to 179.999°. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 120° as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Keep each side opposite its own corner: side a faces angle A, b faces B, c faces C.
What this means: Latitude 1
- What it is
- The latitude of the starting point in decimal degrees: its angle north or south of the equator.
- Units and format
- Enter a number in °: degrees of angle. The form accepts -90° to 90°. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with -33.87° as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- South latitudes are negative. Convert degrees, minutes and seconds to decimal degrees first.
What this means: Longitude 1
- What it is
- The longitude of the starting point in decimal degrees: its angle east or west of the prime meridian. The bearing is measured from this point.
- Units and format
- Enter a number in °: degrees of angle. The form accepts -180° to 180°. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 151.21° as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- West longitudes are negative; east longitudes are positive.
What this means: Latitude 2
- What it is
- The latitude of the destination point in decimal degrees, north positive and south negative.
- Units and format
- Enter a number in °: degrees of angle. The form accepts -90° to 90°. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with -37.81° as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Use the same convention as point 1; mixing signs puts the point in the wrong hemisphere.
What this means: Longitude 2
- What it is
- The longitude of the destination point in decimal degrees, east positive and west negative.
- Units and format
- Enter a number in °: degrees of angle. The form accepts -180° to 180°. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 144.96° as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Swapping latitude and longitude is a common slip; latitude never exceeds 90° either way.
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.
BIZOMAT tools are screening and decision-support utilities. Verify important financial, engineering, safety, legal or operational decisions using appropriate professional standards and your organisation’s approved data.
Open the Spherical Triangle and Great-Circle Distance calculator