# Book:Euclid/The Elements/Book XII

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## Euclid: *The Elements: Book XII*

Published $c. 300 B.C.E$.

### Contents

Book $\text{XII}$: Cones, Pyramids and Cylinders

- Proposition $1$: Areas of Similar Polygons Inscribed in Circles are as Squares on Diameters
- Proposition $2$: Areas of Circles are as Squares on Diameters
- Proposition $3$: Tetrahedron divided into Two Similar Tetrahedra and Two Equal Prisms
- Proposition $4$: Proportion of Sizes of Tetrahedra divided into Two Similar Tetrahedra and Two Equal Prisms
- Proposition $5$: Sizes of Tetrahedra of Same Height are as Bases
- Proposition $6$: Sizes of Pyramids of Same Height with Polygonal Bases are as Bases
- Proposition $7$: Prism on Triangular Base divided into Three Equal Tetrahedra
- Proposition $8$: Volumes of Similar Tetrahedra are in Triplicate Ratio of Corresponding Sides
- Proposition $9$: Tetrahedra are Equal iff Bases are Reciprocally Proportional to Heights
- Proposition $10$: Volume of Cone is Third of Cylinder on Same Base and of Same Height
- Proposition $11$: Volume of Cones or Cylinders of Same Height are in Same Ratio as Bases
- Proposition $12$: Volumes of Similar Cones and Cylinders are in Triplicate Ratio of Diameters of Bases
- Proposition $13$: Volumes of Parts of Cylinder cut by Plane Parallel to Opposite Planes are as Parts of Axis
- Proposition $14$: Volumes of Cones or Cylinders on Equal Bases are in Same Ratio as Heights
- Proposition $15$: Cones or Cylinders are Equal iff Bases are Reciprocally Proportional to Heights
- Proposition $16$: Construction of Equilateral Polygon with Even Number of Sides in Outer of Concentric Circles
- Proposition $17$: Construction of Polyhedron in Outer of Concentric Spheres
- Proposition $18$: Volumes of Spheres are in Triplicate Ratio of Diameters