Sphere - Triangle Area

The Area of a Triangle on a Sphere calculator computes the surface area of a triangle on a sphere based on the angle of the three corners and the radius of the sphere.

INSTRUCTIONS: Choose units and enter the following:

• (α) Angle opposite of Side A
• (β) Angle opposite of Side B
• (γ) Angle opposite of Side C
• (r) Radius of the Sphere (Earth Mean Radius: 6371.009 km)

Area of Triangle on a Sphere (A): The calculator returns the area in square kilometers.  However this can be automatically converted to compatible units via the pull-down menu.

The Math / Science

The formula for the area of a spherical triangle on the surface of a sphere of radius ( r) formed by three great circle arc is:

     A = (α + β + γ - π)⋅r²

where:

• A = area of triangle on surface of a sphere
• α = first angle
• β = second angle
• γ = third angle
• r = radius of the sphere

The triangle is defined by three angles (α,β,γ) at the vertices of the triangle. 

Constraints:

1. The three angles must have a sum greater than π. 
2. Each angle must be greater than zero and less than π.

Triangle Calculators

• Area of a triangle base on the base and height
• Area of a triangle using two sides and the interior angle.
• Area of a triangle using two angles and the interior side.
• Area of a triangle using three known sides.
• Area of a triangle defined by three points (x,y) in a plane.
• Height of a triangle based on the area and base
• Interior Angle of a triangle based on the length of three sides
• Semi-perimeter of a triangle
• Area between two vectors
• Area of Triangle on a Sphere

Sphere Calculators

• Sphere Surface Area based radius (r)
• Sphere Surface Area from Volume
• Sphere Volume from Radius
• Sphere Volume from Circumference
• Sphere Volume from Surface Area
• Sphere Volume from Mass and Density
• Sphere Radius from Volume
• Sphere Radius from Surface Area
• Sphere Weight (Mass) from volume and density
• Sphere Density
• Area of Triangle on a Sphere
• Distance between Two Points on a Sphere
• Sphere Cap Surface Area
• Sphere Cap Volume
• Sphere Cap Weight (Mass) 
• Sphere Segment Volume
• Sphere Segment Weight (Mass)
• Sphere Segment Wall Surface Area (without the circular top and bottom ends)
• Sphere Segment Full Surface Area  (with the top and bottom circles, aka ends)
• Volume of Spherical Shell
• Mass of Spherical Shell