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Spherical Cone — Definition, Formula & Examples

A spherical cone is the solid of revolution created when a circular sector (a 'pie slice' of a circle) is rotated about one of its straight edges (a radius). It combines a cone-like region with a spherical cap on top.

Given a sphere of radius RR and a spherical cap of height hh cut from it, the spherical cone is the solid bounded by the lateral surface of the cone whose apex is the center of the sphere and whose base is the rim of the cap, together with the spherical cap itself. Equivalently, it is the solid generated by revolving a circular sector of radius RR and half-angle θ\theta about its axis of symmetry.

Key Formula

V=23πR2hV = \frac{2}{3}\pi R^2 h
Where:
  • VV = Volume of the spherical cone
  • RR = Radius of the generating sphere
  • hh = Height of the spherical cap

How It Works

To picture a spherical cone, start with a full sphere of radius RR and slice off a spherical cap of height hh. Then connect every point on the cap's circular rim back to the center of the sphere with straight lines. The enclosed solid — part cone, part cap — is the spherical cone. Its volume depends only on RR and hh, and the formula is surprisingly compact.

Worked Example

Problem: A sphere has radius R = 6 cm. A spherical cap of height h = 2 cm is cut from it. Find the volume of the resulting spherical cone.
Write the formula: Use the spherical cone volume formula.
V=23πR2hV = \frac{2}{3}\pi R^2 h
Substitute values: Plug in R = 6 and h = 2.
V=23π(6)2(2)=23π⋅36⋅2V = \frac{2}{3}\pi (6)^2 (2) = \frac{2}{3}\pi \cdot 36 \cdot 2
Simplify: Multiply the constants.
V=144π3=48π≈150.80 cm3V = \frac{144\pi}{3} = 48\pi \approx 150.80 \text{ cm}^3
Answer: The volume of the spherical cone is 48π≈150.8048\pi \approx 150.80 cm³.

Why It Matters

Spherical cones appear in optics and antenna design, where radiation patterns fill cone-shaped regions of a sphere. Understanding their geometry also reinforces how solids of revolution connect 2D sectors to 3D volumes, a key idea in calculus-based volume problems.

Common Mistakes

Mistake: Confusing the cap height h with the sphere's radius R.
Correction: The cap height h is the perpendicular distance from the flat base of the cap to the top of the curved surface, not the full radius. Always check that h ≤ 2R.