Euclid’s Elements — Book XIII, Proposition 15

Constructing a Cube Inside a Sphere

In Book XIII of Euclid’s Elements, Proposition 15 shows how to construct a regular cube and perfectly inscribe it in a given sphere.

Euclid also establishes an elegant relationship between the diameter of the sphere and the edge length of the cube, demonstrating once again the harmony of geometry and proportion.

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Proposition Statement

To construct a cube, inscribe it in a given sphere, and prove that the square on the diameter of the sphere is triple the square on the side of the cube.

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Geometric Setup

1. Begin with a sphere of diameter AB.

2. On AB, choose a point C so that:

   $$AC=2\cdot CB\mathrm{.A}$$

3. Construct a semicircle ADB on AB and draw CD perpendicular to AB.

4. Join DB and construct a square EFGH with side length equal to DB.

5. From each vertex E, F, G, H, erect perpendiculars (EK, FL, GM, HN) to the plane of the square.

6. Set each perpendicular equal to one edge of the square (EF) and join the endpoints to form the top square KLMN.

7. Connect corresponding vertices to form the cube EFGH-KLMN.

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Key Results

1. Construction of the Regular Cube

• By construction, the solid formed has:

  • 6 faces → all perfect squares,

  • 12 edges → all equal,

  • 8 vertices → symmetrically arranged.

• Thus, we have created a regular cube.

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2. The Cube Is Inscribed in the Sphere

• Euclid proves that every vertex of the cube lies on the surface of the sphere.

• This is achieved by showing that diagonals such as KG pass through semicircles related to the construction, ensuring perfect symmetry.

• Thus, the cube is comprehended within the sphere.

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3. Relationship Between the Diameter and the Cube’s Side

Euclid establishes the fundamental identity:

$$A{B}^2=3\cdot E{F}^2$$

where:

• AB = diameter of the sphere,

• EF = side of the cube.

Equivalently:

$$EF = \frac{AB}{\sqrt{3}}$$

Sketch of the Proof

1. Square and Diagonal Properties

   • In square EFGH, the diagonal EG satisfies:

     $$E{G}^2=E{F}^2+F{G}^2=2\cdot E{F}^2\mathrm{.}$$

   • Similarly, the diagonal GK of the cube satisfies:

     $$G{K}^2=E{G}^2+E{K}^2=2\cdot E{F}^2+E{F}^2=3\cdot E{F}^2\mathrm{.}$$

2. Sphere Diameter Equality

   • Euclid shows that GK, the main diagonal of the cube, equals AB, the diameter of the sphere.

3. Final Result

   • Substituting $GK = AB$, we get:

     $$A{B}^2=3\cdot E{F}^2\mathrm{.}$$

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Mathematical Significance

This proposition is crucial for understanding the geometry of the cube within a sphere:

• A regular cube inscribed in a sphere has its main diagonal equal to the sphere’s diameter.

• In modern notation, if the circumsphere radius is $R$ and the cube’s edge is $a$, then:

  $$R=\frac{a\sqrt{3}}{2},$$

  which is equivalent to Euclid’s result.

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Summary

Euclid constructs a regular cube inside a sphere and proves that:

$$\boxed{{\displaystyle A{B}^2=3\cdot E{F}^{\mathrm{2}}}}$$

where:

• AB = diameter of the sphere,

• EF = side of the cube.

Thus, the cube fits perfectly inside the sphere, and its edge length relates beautifully to the sphere’s size.

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Why This Matters

This is the first time Euclid introduces the cube as one of the five Platonic solids in Book XIII.

The result connects 3D geometry with planar constructions and shows how ratios of squares govern the perfect fit of solids inside spheres.

It’s also a foundational step toward understanding more complex solids like the icosahedron and dodecahedron.