Spherical Triangle Sector Calculator

Name: Spherical Triangle Sector - Geometry Calculator
Author: Jürgen Kummer

Calculations for what is referred to here as a spherical triangle sector. This term describes here a shape formed from a specific spherical triangle, with straight edges extending from its vertices to the center of the sphere. This specific spherical triangle features two right angles and two sides of equal length, making it isosceles. If the sphere is viewed as a globe, both right angles and the side connecting them lie along the equator. This side serves as the base of the isosceles spherical triangle. The two legs lie along lines of longitude and meet at a pole, forming an angle α. If α is also a right angle, this shape is a spherical corner.
Enter the radius of the sphere and the base length of the spherical triangle. Choose the number of decimal places, then click Calculate. Please enter the angle in degrees, here you can convert angle units.

Formulas:

b = \frac{πr}{2}

α = \frac{a}{2 \cdot π \cdot r} \cdot 360 °

α in rad:

S = α \cdot r^{2}

A = S + \frac{π + α}{2} \cdot r^{2}

V = \frac{αr^{3}}{3}

pi:

π = 3.141592653589793...

Radiuses and lengths have the same unit (e.g. meter), the areas have this unit squared (e.g. square meter), the volume has this unit to the power of three (e.g. cubic meter). A/V has this unit ^-1.

An isosceles spherical triangle sector is a halved spherical wedge. The curved part of the sector's surface is a spherical triangle. This belongs to spherical geometry, not to Euclidean geometry. However, a spherical triangle sector is perfectly compatible with Euclidean geometry, provided the curved part of the surface is not referred to as a triangle.
The total surface area of the spherical triangle sector consists of the spherical triangle plus three circular sectors, two of which are identical. See the calculator there for the angles between the respective lines. The volume of the spherical triangle sector has the same ratio to the sphere's volume as the area of the spherical triangle has to the surface area of the whole sphere. The same approach can be used to calculate the corresponding volume beneath a general spherical triangle.

Calculation of volume and other dimensions: spherical triangle sector

Last updated on 09/22/2026. Author: Jürgen Kummer

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