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Spherical Cap — Definition, Formula & Examples

A spherical cap is the region of a sphere that lies above (or below) a plane that intersects the sphere. Think of slicing the top off an orange — the piece you remove is shaped like a spherical cap.

Given a sphere of radius RR and a plane intersecting it, a spherical cap is the solid bounded by the spherical surface on one side of the plane and the disk formed by the intersection. The height hh of the cap is the perpendicular distance from the cutting plane to the nearest pole of the sphere.

Key Formula

V=πh23(3R−h)Acurved=2πRhV = \frac{\pi h^2}{3}(3R - h) \qquad A_{\text{curved}} = 2\pi R h
Where:
  • VV = Volume of the spherical cap
  • AcurvedA_{\text{curved}} = Area of the curved (spherical) surface of the cap
  • RR = Radius of the original sphere
  • hh = Height of the cap (distance from the cutting plane to the top of the cap)

How It Works

A spherical cap is fully determined by two parameters: the sphere's radius RR and the cap's height hh. From these, you can derive the base radius of the cap as a=2Rh−h2a = \sqrt{2Rh - h^2}. The volume and curved surface area each have clean closed-form expressions in terms of RR and hh. When h=Rh = R, the cap is exactly a hemisphere. When h=2Rh = 2R, the cap is the entire sphere.

Worked Example

Problem: A sphere has radius R = 10 cm. A plane cuts the sphere at a height h = 4 cm from the top. Find the volume and the curved surface area of the resulting spherical cap.
Volume: Apply the volume formula with R = 10 and h = 4.
V=π(4)23(3⋅10−4)=16π3(26)=416π3≈435.6 cm3V = \frac{\pi (4)^2}{3}(3 \cdot 10 - 4) = \frac{16\pi}{3}(26) = \frac{416\pi}{3} \approx 435.6 \text{ cm}^3
Curved Surface Area: Apply the curved surface area formula.
Acurved=2π(10)(4)=80π≈251.3 cm2A_{\text{curved}} = 2\pi(10)(4) = 80\pi \approx 251.3 \text{ cm}^2
Base Radius (bonus): The radius of the circular base of the cap can be found for reference.
a=2(10)(4)−42=80−16=64=8 cma = \sqrt{2(10)(4) - 4^2} = \sqrt{80 - 16} = \sqrt{64} = 8 \text{ cm}
Answer: The spherical cap has volume 416π3≈435.6\frac{416\pi}{3} \approx 435.6 cm³ and curved surface area 80π≈251.380\pi \approx 251.3 cm².

Why It Matters

Spherical caps appear in optics (modeling lens surfaces), civil engineering (domed roofs and tanks), and geodesy (calculating areas on Earth's surface). In multivariable calculus, deriving the cap formulas is a standard exercise in integration with spherical coordinates.

Common Mistakes

Mistake: Using hh as the distance from the center of the sphere to the cutting plane instead of the cap's actual height.
Correction: The cap height hh is measured from the cutting plane to the top of the cap, not from the sphere's center. If dd is the distance from the center to the plane, then h=R−dh = R - d.