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Spherical Wedge — Definition, Formula & Examples

A spherical wedge is the 3D region of a sphere enclosed between two half-planes that meet along a diameter. Think of it as a thick slice of an orange, where the blade cuts from the center outward on two sides.

Given a sphere of radius rr and a dihedral angle θ\theta (in radians) formed by two half-planes sharing a common diameter, the spherical wedge (also called an ungula) is the solid region bounded by the sphere's surface and the two half-planes.

Key Formula

V=23 r3 θV = \frac{2}{3}\,r^3\,\theta
Where:
  • VV = Volume of the spherical wedge
  • rr = Radius of the sphere
  • θ\theta = Dihedral angle between the two half-planes, in radians

How It Works

A spherical wedge is defined by two ingredients: the sphere's radius rr and the dihedral angle θ\theta between the two cutting planes. Since the wedge is a constant-angle fraction of the full sphere, its volume scales linearly with θ\theta. When θ=2π\theta = 2\pi, the wedge is the entire sphere, recovering 43πr3\frac{4}{3}\pi r^3. The surface of the wedge that lies on the sphere itself is called a spherical lune.

Worked Example

Problem: Find the volume of a spherical wedge cut from a sphere of radius 6 cm by a dihedral angle of 60°.
Convert the angle to radians: Since 60° equals π/3 radians:
θ=π3\theta = \frac{\pi}{3}
Apply the volume formula: Substitute r = 6 and θ = π/3 into the formula:
V=23(6)3(π3)=23(216)(π3)=432π9=48πV = \frac{2}{3}(6)^3\left(\frac{\pi}{3}\right) = \frac{2}{3}(216)\left(\frac{\pi}{3}\right) = \frac{432\pi}{9} = 48\pi
Compute the numerical result: Evaluate to get the approximate volume:
V≈150.80 cm3V \approx 150.80 \text{ cm}^3
Answer: The volume of the spherical wedge is 48π≈150.8048\pi \approx 150.80 cm³.

Why It Matters

Spherical wedges appear in multivariable calculus when setting up triple integrals in spherical coordinates, since integration over an angle range θ naturally defines a wedge. They also arise in geophysics and astronomy when modeling sectors of planets or celestial spheres.

Common Mistakes

Mistake: Using the angle in degrees directly in the formula instead of converting to radians.
Correction: The formula V=23r3θV = \frac{2}{3}r^3\theta requires θ\theta in radians. Always convert first: multiply degrees by π180\frac{\pi}{180}.