Sphere

A sphere is a perfectly round three-dimensional object. Every point on its surface is at the same distance (radius) from its center.

r

Properties

A sphere is a 3D analog of the circle. It is the set of all points in three-dimensional space at a fixed distance (radius) from a center point. The sphere has the smallest surface area for a given volume of any shape.

Basic Formulas

V = ⁴⁄₃πr³ Volume of sphere
A = 4πr² Surface area
V = πd³/6 Volume in terms of diameter d
A = πd² Surface area in terms of diameter

Hemisphere

For a hemisphere (half of a sphere):

V_hemi = ²⁄₃πr³ Volume of hemisphere
A_hemi = 3πr² Surface area (including base)

Example

Given: radius r = 6 cm

Volume: V = (4/3) · π · 6³ = (4/3) · π · 216 = 288π ≈ 904.779 cm³

Surface area: A = 4 · π · 6² = 4 · π · 36 = 144π ≈ 452.389 cm²

Hemisphere volume: V_hemi = (2/3) · π · 216 = 144π ≈ 452.389 cm³

Notes

The Earth is approximately a sphere (actually an oblate spheroid). The sphere is the most volume-efficient shape — it contains the maximum volume for a given surface area. This is why soap bubbles form spheres.

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