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REGULAR POLYHEDRONS.

673. DEF. A regular polyhedron is a polyhedron whose faces are equal regular polygons, and whose polyhedral angles are equal.

PROPOSITION XXIX. PROBLEM.

674. To determine the number of regular convex polyhedrons possible.

A convex polyhedral angle must have at least three faces, and the sum of its face angles must be less than 360° (§ 581).

1. Since each angle of an equilateral triangle is 60°, convex polyhedral angles may be formed by combining three, four, or five equilateral triangles. The sum of six such angles is 360°, and hence greater than the sum of the face angles of a convex polyhedral angle. Hence, three regular convex polyhedrons are possible with equilateral triangles for faces.

2. Since each angle of a square is 90°, a convex polyhedral angle may be formed by combining three squares. The sum of four such angles is 360°, and therefore greater than the sum of the face angles of a convex polyhedral angle. Hence, one regular convex polyhedron is possible with squares.

3. Since each angle of a regular pentagon is 108° (§ 206), a convex polyhedral angle may be formed by combining three regular pentagons. The sum of four such angles is 432°, and therefore greater than the sum of the face angles of a convex polyhedral angle. Hence, one regular convex polyhedron is possible with regular pentagons.

4. The sum of three angles of a regular hexagon is 360°, of a regular heptagon is greater than 360°, etc. Hence, only five regular convex polyhedrons are possible.

The five regular polyhedrons are called the tetrahedron, the hexahedron, the octahedron, the dodecahedron, the icosahedron.

Q. E. F.

675. The regular polyhedrons may be constructed as follows: the Draw the diagrams given below on stiff paper. Cut through the full lines and fold on the dotted lines. Bring the edges together so as to form the respective polyhedrons, and keep the edges in contact by pasting strips or laps of paper, as shown in the diagrams.

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Tetrahedron. Hexahedron. Octahedron. Dodecahedron. Icosahedron.

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CYLINDERS.

676. DEF. A cylindrical surface is a curved surface generated by a straight line, which moves parallel to a fixed straight line and constantly touches a fixed curve not in the plane of the straight line.

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The moving line is called the generatrix, and the fixed, curve the directrix.

677. DEF. The genera

trix in any position is

Cylindrical Surface.

called an element of the cylindrical surface.

678. DEF. A cylinder is a solid bounded by a cylindrical surface and two parallel plane surfaces.

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Right Cylinder.

679. DEF. The two plane surfaces are called the bases, and the cylindrical surface is called the lateral surface.

680. DEF. The altitude of a cylinder is the perpendicular distance between the planes of its bases. The elements of a cylinder are all equal.

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it may be generated by the revolution of a rectangle about

Inscribed Prism.

one side as an axis.

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684. DEF. Similar cylinders of revolution are cylinders generated. by the revolution of similar rectangles about homologous sides.

685. DEF. A tangent line to a cylinder is a straight line, not an element, which touches the lateral surface of the cylinder but does not intersect it.

686. DEF. A tangent plane to a cylinder is a plane which con

tains an element of the cylinder but does not cut the surface. The element contained by the plane is called the element of

contact.

687. DEF. A prism is inscribed in a cylinder when its lateral edges are elements of the cylinder and its bases are inscribed in the bases of the cylinder.

688. DEF. A prism is circumscribed about a cylinder when its lateral edges are parallel to the elements of the cylinder and its bases are circumscribed about the bases of the cylinder.

Circumscribed Prism.

689. DEF. A section of a cylinder is the figure formed by its intersection with a plane passing through it.

A right section of a cylinder is a section made by a plane perpendicular to its elements.

See

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PROPOSITION XXX. THEOREM.

690. Every section of a cylinder made by a plane passing through an element is a parallelogram.

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Let ABCD be a section of the cylinder AC made by a plane passing through the element AD.

To prove that ABCD is a parallelogram.

Proof. Through B draw a line in the plane AC, I to AD. This line is an element of the cylindrical surface. § 676 Since this line is in both the plane and the cylindrical surface, it must be their intersection and coincide with BC. Hence, BC coincides with a straight line parallel to AD. Therefore, BC is a straight line to AD.

Also AB is a straight line I to CD. .. ABCD is a parallelogram.

§ 528

§ 166

Q.E. D.

691. COR. Every section of a right cylinder made by a

plane passing through an element is a rectangle.

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