Ex. 797. What is the altitude of the frustum of a regular hexagonal pyramid, if its volume is 16cum and the sides of its bases are respectively 1.5m and 2.5m?

Ex. 798. A pyramid 20 ft. high has 100 sq. ft. in its base; a section parallel to the base has an area of 55 sq. ft. How far is the section from the base?

Ex. 799. What is the volume of an oblique truncated triangular prism whose edges are 5m, 7m, and 9m, and the area of whose right section is 16sq m ? Ex. 800. What is the edge of a cube whose entire area is 18q m ?

Ex. 801. The base of a pyramid contains 121 sq. ft.; a section parallel to the base and 3 ft. from the vertex contains 49 sq. ft. What is the altitude of the pyramid ?

Ex. 802. What is the lateral area of a regular hexagonal pyramid whose base is inscribed in a circle whose diameter is 15 ft., the altitude of the pyramid being 8 ft.? What is the volume of the pyramid ?

Ex. 803. Any lateral edge of a right prism is equal to the altitude.

Ex. 804. The square of a diagonal of a rectangular parallelopiped is equal to the sum of the squares of its three edges.

Ex. 805. If the edges of a tetrahedron are all equal, the sum of the angles at any corner is equal to two right angles.

Ex. 806. The section of a triangular pyramid made by a plane parallel to two opposite edges is a parallelogram.

Ex. 807. The lateral faces of right prisms are rectangles.

Ex. 808. The section of a prism made by a plane parallel to a lateral edge is a parallelogram.

Ex. 809. The diagonal of a cube is equal to the product of its edge and √3.

Ex. 810. The volume of a regular prism is equal to the product of its lateral area and one half the apothem of the base.

Ex. 811. Any straight line passing through the center* of a parallelopiped and terminated by two faces is bisected at the center.

Ex. 812. If any two non-parallel diagonal planes of a prism are perpendicular to the base, the prism is a right prism.

Ex. 813. The base of a pyramid is 16 sq. ft. and its altitude is 7 ft. What is the area of a section parallel to the base, if it is 2 ft. 6 in. from the base?

Ex. 814. The edges of a rectangular parallelopiped are 3 in., 4 in., and 6 in. What is the area of its diagonal planes and the length of its diagonal line?

*The center of a parallelopiped is the intersection of its diagonals.

SOLID GEOMETRY - BOOK VIII.

A portion of a railway embankment is 18 ft. by 380 ft. at the ft. by 380 ft. at the bottom. If its height is 12 ft., how many or loads, of earth does it contain ?

If the four diagonals of a four-sided prism pass through a comthe prism is a parallelopiped.

If a pyramid is cut by a plane parallel to its base, the pyramid milar to the given pyramid.

The lateral area of a right prism is less than the lateral area of prism having the same base and altitude.

If a section of a pyramid made by a plane parallel to the base altitude, the area of the section is one fourth the area of the base, amid cut off is one eighth of the original pyramid.

The volume of a triangular prism is equal to one half the any lateral face by its distance from the opposite edge.

If the diagonals of three unequal faces of a rectangular parale given, compute the edges.

What is the lateral area of a regular pyramid whose slant Oft., the base being a pentagon inscribed in a circle whose radius hat is the volume ?

The volume of a rectangular parallelopiped is 336cum, its total m, and its altitude is 4m. What are the dimensions of its base ?

A pyramid weighs 30Kg, and its altitude is 12dm. A plane he base cuts off a frustum which weighs 15Kg. What is the altifrustum?

Each side of the base of a regular triangular pyramid is 3 in., ude is 8 in. What are its lateral edge and lateral area?

The volume of a regular tetrahedron is equal to the product of its edge and√2.

The volume of a regular octahedron is equal to the product of its edge and √2.

. Any plane passing through the center of a parallelopiped to two equal solids.

The lateral area of a regular pyramid is greater than its base.

The lateral edge of the frustum of a regular triangular pyramid side of one base is 5 ft., and of the other 4 ft. What is the

The sum of the perpendiculars from any point within a regular to each of its four faces is equal to its altitude.

In a regular tetrahedron an altitude is equal to three times the

BOOK IX

CYLINDERS AND CONES

584. A surface, generated by a moving straight line which always remains parallel to its original posi

tion and continually touches a given curved A

line, is called a Cylindrical Surface.

The moving straight line is called the gen

eratrix, and the given curved line is called the directrix.

The generatrix in any position is called.

an element of the surface.

B

F

585. A solid bounded by a cylindrical surface and two parallel planes which cut all its elements is called a Cylinder.

The plane surfaces are called the bases and the cylindrical surface is called the lateral, or convex surface of the cylinder.

All elements of a cylinder are equal. § 464.

The perpendicular distance between its bases is the altitude of the cylinder.

586. A section of a cylinder made by a plane perpendicular to its elements is called a Right Section.

587. A cylinder whose elements are perpendicular to its base is called a Right Cylinder.

588. A cylinder whose elements are not perpendicular to its base is called an Oblique Cylinder.

589. A cylinder whose bases are circles is a Circular Cylinder. The straight line joining the centers of the bases of a circular cylinder is called the axis of the cylinder.

590. A right circular cylinder is called a Cylinder of Revolution, because it may be generated by the revolution of a rectangle about one of its sides.

Cylinders of revolution generated by similar rectangles revolving about homologous sides are similar.

591. A plane which contains an element of a cylinder and does not cut the surface is a Tangent Plane to the cylinder.

The element is called the element of contact.

592. Any straight line that lies in a tangent plane and cuts the element of contact is a Tangent Line to the cylinder.

593. When the bases of a prism are inscribed in the bases of a cylinder and its lateral edges are elements of the cylinder, the prism is said to be inscribed in the cylinder.

594. When the bases of a prism are circumscribed about the bases of a cylinder and its lateral edges are parallel to the elements of the cylinder, the prism is said to be circumscribed about the cylinder.

Proposition I

595. 1. Form a cylinder and cut it by any plane through an element of its surface (§ 519 N.). What plane figure is the section made by the cutting plane?

2. If the cylinder is a right cylinder, what plane figure does such a plane make?

Theorem. Any section of a cylinder made by a plane passing through an element is a parallelogram.

Data: Any section of the cylinder EF, as

ABCD, made by a plane passing through AB, an element of the surface.

To prove ABCD a parallelogram.

Proof. The plane passing through the element AB cuts the circumference of the base in a second point, as D. Draw DC AB. Then, § 63, DC is in the plane BAD; and, § 584,

E

DC is an element of the cylinder.

B

Hence, DC, being common to the plane and the lateral surface of the cylinder, is their intersection.

596. Cor. Any section of a right cylinder made by a plane passing through an element is a rectangle.

Proposition II

597. 1. Form a cylinder. How do its bases compare?

2. Cut the cylinder by parallel planes which cut all its elements. How do the sections thus made compare with each other?

3. How does a section made by a plane parallel to the base compare with the base?

Theorem. The bases of a cylinder are equal.

Data: Any cylinder, as MG, whose bases are HG and MN.

Proof. Take any three points in the perimeter of the upper base, as D, E, F, and from them draw the elements of the surface DA, EB, FC, respectively.

Draw AB, BC, AC, DE, EF, and DF.

§§ 585, 584, AD, BE, and CF are equal and parallel;

.. § 150, AE, AF, and BF are parallelograms;

Why?

Apply the upper base to the lower base so that DE shall fall upon AB.

But F is any point in the perimeter of the upper base, therefore, every point in the perimeter of the upper base will fall upon the perimeter of the lower base.

Hence, § 36,

HG MN.

Therefore, etc.

Q.E.D.

598. Cor. I. The sections of a cylinder made by parallel planes cutting all its elements are equal.

599. Cor. II. The axis of a circular cylinder passes through the centers of all the sections parallel to the bases.