Spherical Geometry Formulas & 3D Shapes Calculator

Base Geometry 🌐

A perfectly symmetrical 3D round object where every exterior surface point is equidistant from the center point.

V = ⁴⁄₃ \times π \times r³

A = 4 \times π \times r²

Half Sphere 🌓

Exactly half of a full sphere formed by a plane passing straight through the sphere's geometric center.

V = ²⁄₃ \times π \times r³

Total A = 3 \times π \times r²

Segment 🧢

The portion of a sphere cut off by any intersecting plane at height \(h\) from the pole.

V = ⅓ \times π \times h² \times (3r - h)

Curved A = 2 \times π \times r \times h

Shell ⚪

The spatial region bounded between two concentric spheres with outer radius \(R\) and inner radius \(r\).

V = ⁴⁄₃ \times π \times (R³ - r³)

Total A = 4π(R² + r²)

Cone + Cap 🍦

A solid formed by a spherical cap and the cone bounded by the sphere center and the cap's base boundary.

V = ²⁄₃ \times π \times r² \times h

Total A = πr(2h + a)

Stretched Sphere 🏉

A 3D closed quadric surface that is the 3D analogue of an ellipse with semi-principal axes \(a\), \(b\), and \(c\).

V = ⁴⁄₃ \times π \times a \times b \times c

A \approx 4π ((a^p b^p + ...)/3)^(1/p)

Spherical Geometries Quick Comparison

Summary of volumetric equations and surface area expressions across all shapes.

3D Shape	Volume Formula (\(V\))	Surface Area Formula (\(A\))	Required Variables
Sphere	V = ⁴⁄₃πr³	A = 4πr²	Radius \(r\)
Hemisphere	V = ²⁄₃πr³	A = 3πr² (total)	Radius \(r\)
Spherical Cap	V = ⅓πh²(3r − h)	A = 2πrh (curved)	Radius \(r\), Height \(h\)
Spherical Shell	V = ⁴⁄₃π(R³ − r³)	A = 4π(R² + r²)	Outer \(R\), Inner \(r\)
Spherical Sector	V = ²⁄₃πr²h	A = πr(2h + a)	Radius \(r\), Height \(h\)
Ellipsoid	V = ⁴⁄₃πabc	Knudsen / Thomsen approx	Semi-axes \(a, b, c\)