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Spherical Triangle — Definition, Formula & Examples

A spherical triangle is a figure on the surface of a sphere formed by three arcs of great circles that intersect pairwise at three distinct vertices. Unlike a plane triangle, its interior angles always sum to more than 180°.

A spherical triangle on a sphere of radius rr is a region bounded by three great-circle arcs connecting three non-collinear points on the sphere, where each arc subtends a central angle less than π\pi. The angular excess E=(α+β+γ)−πE = (\alpha + \beta + \gamma) - \pi is strictly positive, and the area of the triangle equals E⋅r2E \cdot r^2.

Key Formula

A=(α+β+γ−π) r2A = (\alpha + \beta + \gamma - \pi)\, r^2
Where:
  • AA = Area of the spherical triangle
  • α,β,γ\alpha, \beta, \gamma = Interior angles of the triangle (in radians)
  • rr = Radius of the sphere
  • π\pi = The constant pi (≈ 3.14159)

How It Works

Each side of a spherical triangle is an arc of a great circle, which is the intersection of the sphere with a plane through its center. Sides are measured as angles (in radians) subtended at the center, not as linear distances. The angle at each vertex is the dihedral angle between the two planes defining the adjacent sides. Because the surface curves, the three interior angles α\alpha, β\beta, γ\gamma always add up to strictly more than π\pi radians (180°). The amount by which the sum exceeds π\pi is called the spherical excess EE, and it directly determines the triangle's area via A=Er2A = E r^2.

Worked Example

Problem: A spherical triangle on a sphere of radius 10 has interior angles of 90°, 80°, and 70°. Find its area.
Step 1: Convert the angles to radians.
α=π2,β=4π9,γ=7π18\alpha = \frac{\pi}{2},\quad \beta = \frac{4\pi}{9},\quad \gamma = \frac{7\pi}{18}
Step 2: Compute the angle sum and the spherical excess E.
α+β+γ=π2+4π9+7π18=9π+8π+7π18=24π18=4π3\alpha + \beta + \gamma = \frac{\pi}{2} + \frac{4\pi}{9} + \frac{7\pi}{18} = \frac{9\pi + 8\pi + 7\pi}{18} = \frac{24\pi}{18} = \frac{4\pi}{3}
Step 3: Subtract π to get the excess, then multiply by r².
E=4π3−π=π3,A=π3⋅102=100π3≈104.7E = \frac{4\pi}{3} - \pi = \frac{\pi}{3}, \qquad A = \frac{\pi}{3} \cdot 10^2 = \frac{100\pi}{3} \approx 104.7
Answer: The area of the spherical triangle is 100π3≈104.7\dfrac{100\pi}{3} \approx 104.7 square units.

Why It Matters

Spherical triangles are essential in navigation and astronomy, where positions on Earth or the celestial sphere are connected by great-circle routes. They also appear in geodesy, satellite communication, and crystallography whenever calculations must account for the curvature of a sphere.

Common Mistakes

Mistake: Assuming the angle sum is exactly 180° as in a plane triangle.
Correction: On a sphere the angles always sum to more than 180°. The excess is what determines the triangle's area and distinguishes spherical geometry from Euclidean geometry.