Solid geometry: fundamentals

Solid geometry studies figures in 3D space. From pool volume calculation to silo design, from bottle capacity to ball sizing, volume and surface formulas are fundamental tools in engineering, architecture, physics, and daily life.

The 9 shapes covered

1. Sphere
2. Cube
3. Rectangular prism (box)
4. Cylinder
5. Cone
6. Square pyramid
7. Triangular prism
8. Hemisphere
9. Torus (donut)

Core formulas

1. Sphere

Most symmetric solid:

V = (4/3)·π·r³

S = 4·π·r²

Unique property: of all solids with given volume, sphere minimizes surface area. That's why soap bubbles are spherical.

2. Cube

V = a³, S = 6·a²

Space diagonal: d = a·√3.

3. Rectangular prism

V = l·w·h, S = 2·(lw + wh + lh)

Diagonal: d = √(l² + w² + h²).

4. Cylinder

V = π·r²·h, S = 2·π·r² + 2·π·r·h

Lateral surface unrolls to rectangle: C × h = 2πr × h.

5. Right cone

V = (1/3)·π·r²·h (one-third of cylinder with same base and height!)

S = π·r² + π·r·ℓ

where ℓ = √(r² + h²) is slant height. Archimedes discovered the 1/3 ratio in the 3rd century BCE.

6. Square pyramid

V = (1/3)·a²·h, S = a² + 2·a·ℓ where ℓ = √((a/2)² + h²)

Egyptian pyramids at Giza are approximately square-based pyramids.

7. Triangular prism

V = (1/2)·b·h_base·L

Typical shape of tents and gabled roofs.

8. Hemisphere

V = (2/3)·π·r³, S = 3·π·r² (dome + base circle)

9. Torus (donut)

Two radii: R (major, donut center to axis), r (minor, tube).

V = 2·π²·R·r², S = 4·π²·R·r

Surprising: both formulas contain π² instead of π.

Relations between solids

Cavalieri's principle

Two solids with same cross-sectional area at every height have same volume. Proved by Bonaventura Cavalieri (1635), precursor to integral calculus.

Cylinder-sphere-cone inscribed

Considering cylinder, sphere, and cone inscribed in cube of side 2r with height 2r:

• V_cylinder = 2π·r³
• V_sphere = (4/3)π·r³
• V_cone = (2/3)π·r³

Ratio: cylinder : sphere : cone = 3 : 2 : 1. Discovered by Archimedes (~240 BCE) — his tomb was decorated with this figure, considered his greatest result.

Euler's polyhedron formula

For any convex polyhedron: V − E + F = 2

• Cube: 8 − 12 + 6 = 2
• Tetrahedron: 4 − 6 + 4 = 2
• Dodecahedron: 20 − 30 + 12 = 2

Practical examples

Soccer ball

22 cm diameter (11 cm radius): V = (4/3)π × 1331 ≈ 5,575 cm³ ≈ 5.6 liters.

Cylindrical tank

Radius 1 m, height 3 m: V = π × 3 ≈ 9,425 liters.

Ice cream cone

Radius 3 cm, height 10 cm: V = (1/3)π × 9 × 10 ≈ 94.25 cm³.

Shipping box

60 × 40 × 30 cm: V = 72,000 cm³ = 72 liters.

Hemispherical pool

Radius 3 m: V = (2/3)π × 27 ≈ 56.55 m³ = 56,550 liters.

Sphere optimal properties

The sphere is optimal in many senses:

• Max volume/surface ratio: for given volume, sphere has smallest surface (3D isoperimetric)
• Max resistance to uniform external pressure: reason for mercury drops, bubbles, pressurized balls
• Minimum free fall drag: spherical symmetry gives predictable fluid behavior
• Equilibrium temperature: planets and stars are spherical by self-gravitation

Other notable solids

The 5 Platonic solids

Only convex polyhedra with congruent faces:

Plato associated them with elements: fire (tetrahedron), earth (cube), air (octahedron), universe (dodecahedron), water (icosahedron). Euclid proved in Elements Book XIII that only these five exist.

Archimedean solids

13 semi-regular polyhedra with faces of multiple types but identical vertices. Archimedes described all; book lost; reconstructed by Kepler 1619.

Volumes in nature

Planets

• Earth: V ≈ 1.08 × 10¹² km³, S ≈ 510 million km²
• Sun: V ≈ 1.41 × 10¹⁸ km³ (1.3 million times Earth)
• Jupiter: V ≈ 1.43 × 10¹⁵ km³

Why planets are spherical

Above critical mass (~500 km radius), self-gravitation crushes body into sphere ("hydrostatic equilibrium"). Below, asteroids keep irregular shape.

Cells and viruses

• Typical human cell (spheroid): V ≈ 4,000 μm³
• E. coli bacterium (cylinder): V ≈ 1-3 μm³
• Virus (icosahedral): V ≈ 10⁻³ - 10⁻⁴ μm³

Solids in art and architecture

• Great Pyramid of Giza: square pyramid, 146 m original height
• Pantheon Rome: hemispherical dome, 21.6 m radius
• Hagia Sophia: dome on pendentives, 32 m diameter
• Guggenheim Museum NY: Frank Lloyd Wright spiral
• 30 St Mary Axe (Gherkin London): ellipsoid of revolution

Volume and capacity units

• 1 m³ = 1,000 liters
• 1 cm³ = 1 ml
• 1 US gallon = 3.785 liters
• 1 UK gallon = 4.546 liters
• 1 US fluid ounce = 29.57 ml
• 1 UK pint = 568 ml (different from US!)

Square-cube law

Doubling radius:

• Surface: × 4 (quadratic)
• Volume: × 8 (cubic)

Consequences:

• Babies lose heat faster (more surface per mass)
• Whales can't live on land (crushed by own weight)
• Giant insects impossible (breathing wouldn't scale)

History

• ~300 BCE Euclid: books XI-XIII of Elements
• ~250 BCE Archimedes: cylinder-sphere-cone ratios
• 1500 Leonardo da Vinci: Platonic solids illustrations
• 1619 Kepler: Harmonices Mundi
• 1635 Cavalieri: principle of indivisibles
• 1750 Euler: V − E + F = 2 polyhedron formula

Excel / Sheets / Python functions

• =PI() * r^2 * h: cylinder volume
• =(4/3) * PI() * r^3: sphere volume
• =(1/3) * PI() * r^2 * h: cone volume

Python: math.pi, math.sqrt, ** operators.

FAQ

Why are bubbles spherical?

Sphere minimizes surface for given volume. Liquid surface tension pushes bubble to minimum-area shape → sphere.

Cone and cylinder: why 1/3?

Proved by Eudoxus (~370 BCE) via exhaustion, rigorously by Archimedes. Today immediate via integral calculus.

Why does torus have π²?

Torus is surface generated by circle (π factor) rotated around axis (2π factor). Product: 2π².

Volume vs capacity?

Conceptually equal. Volume is geometric property (space occupied), capacity is fluid contained. Same units.

Volume of irregular shapes?

Two approaches: decompose into regular solids; Archimedes' method (displacement) — water immersion.

Updated July 2026