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What three-dimensional topics are covered in Books XI through XIII of Euclid’s Elements?

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Three theorems of congruent triangles — Euclidean geometry This diagram depicts three basic theorems that triangles are congruent—that is, they are of equal shape and size—if: (1) two sides and the included angle are equal, designated SAS; (2) two angles and the included side are equal, designated ASA; or (3) all three sides are equal, designated SSS.

In Euclid’s Elements, Books XI through XIII focus on three-dimensional geometry, referred to in Greek as stereometria. The topics covered in these books are as follows:1

Book XI: This book establishes the foundations of solid geometry, containing theorems regarding the intersections of planes, the relationships between lines and planes, and the volumes of parallelepipeds (solids with parallel parallelograms as opposite faces).21
Book XII: This book applies the method of exhaustion—a technique introduced by Eudoxus—to determine the volumes of various solid figures, including the sphere, and to prove that the areas of circles are proportional to the squares of their diameters.21
Book XIII: Serving as a three-dimensional analogue to Book IV, this final book culminates in the construction of the five regular Platonic solids (the pyramid, cube, octahedron, dodecahedron, and icosahedron) within a given sphere.21