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Spherical Sector — Definition, Formula & Examples

A spherical sector is a 3D solid formed by rotating a sector of a circle about an axis of the sphere. It consists of a cone whose apex is at the center of the sphere together with the spherical cap it subtends.

Given a sphere of radius RR, a spherical sector is the solid bounded by a conical surface with its vertex at the center of the sphere and the spherical cap intercepted by that cone. If the cap has height hh, the volume of the spherical sector is 23πR2h\frac{2}{3}\pi R^{2}h.

Key Formula

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

How It Works

Picture slicing an orange wedge from the center outward — the piece you get (rind plus the fleshy interior cone) is shaped like a spherical sector. To find its volume, you only need two measurements: the sphere's radius RR and the height hh of the spherical cap. The cap height hh is the perpendicular distance from the flat base of the cap to the top of the curved surface. Plug both values into the formula V=23πR2hV = \frac{2}{3}\pi R^{2}h and compute.

Worked Example

Problem: A sphere has radius 6 cm. A spherical sector is cut so that its cap has a height of 2 cm. Find the volume of the sector.
Identify values: The sphere's radius is R=6R = 6 cm and the cap height is h=2h = 2 cm.
Apply the formula: Substitute into the volume formula.
V=23π(6)2(2)=23π⋅36⋅2=144π3=48πV = \frac{2}{3}\pi (6)^{2}(2) = \frac{2}{3}\pi \cdot 36 \cdot 2 = \frac{144\pi}{3} = 48\pi
Compute: Evaluate the numerical result.
V≈150.80 cm3V \approx 150.80 \text{ cm}^{3}
Answer: The volume of the spherical sector is 48π≈150.8048\pi \approx 150.80 cm³.

Why It Matters

Spherical sectors appear in engineering when modeling dome-shaped tanks, radar coverage zones, and spotlight beam volumes. Understanding this formula also builds toward multivariable calculus, where you integrate over spherical coordinates to find volumes of revolution.

Common Mistakes

Mistake: Confusing the cap height hh with the full radius RR and substituting RR for hh.
Correction: The cap height hh is only the vertical distance from the base of the cap to the sphere's surface, not the sphere's radius. These are equal only when the cap is a full hemisphere.