Proposition 260 Theorem

The section of a pyramid made by a plane parallel to the base is a polygon similar to the base.

Hypothesis. In the pyramid A-BCDEF, G-M is a section made by a plane to the base B-E.

Consult Prop. 214, Prop. 222, and Prop. 79, Cor. III.

COR. The sections of a pyramid made by two parallel planes which cut all the lateral edges are similar polygons.

If a pyramid is cut by a plane parallel to its base, the lateral edges and the altitude are divided proportionally.

Consult Prop. 227, Cor.

Ex. 1089. If a plane divides the lateral edges of a pyramid proportionally, it is parallel to the base of the pyramid.

If a pyramid is cut by a plane parallel to its base, the area of the section is to the area of the base as the square of its distance from the vertex is to the square of the altitude of the prism.

Consult Prop. 155 and Prop. 261.

COR.

The areas of two parallel sections of a pyramid are to each other as the squares of their distances from the vertex.

If two pyramids having equal altitudes are cut by planes parallel to their bases and at equal distances from their vertices, the sections are to each other as their bases.

Consult Prop. 262.

COR. If two pyramids have equal altitudes and equivalent bases, sections made by planes parallel to their bases and at equal distances from their vertices are equivalent.

Proposition 264 Theorem

The lateral area of a regular pyramid is equal to one half the product of the perimeter of its base and its slant height.

Consult Prop. 151 and Ax. 4.

Let S denote the lateral area, p the perimeter of the base, and the slant height of a regular pyramid. Then

S = pxl.

Ex. 1090. The lateral area of any pyramid is greater than the area of the base.

The lateral area of a frustum of a regular pyramid is equal to one half the product of the sum of the perimeters of its bases and its slant height.

Consult Prop. 152 and Ax. 4.

Let S denote the lateral area, p and p' the perimeters of the bases, and the slant height of a frustum of a regular pyramid. Then

S = { (p + p') × l.

DEFINITIONS. If the altitude of a pyramid is divided into equal parts, and through the points of division planes are passed parallel to the base of the pyramid and on the sections made by these planes as upper bases prisms are constructed having their altitudes equal to one of the equal parts of the altitude of the pyramid and their edges parallel to an edge of the pyramid, the prisms lie wholly within the pyramid and are said to be inscribed in the pyramid. Prisms similarly constructed on the sections as lower bases lie partly without the pyramid and are said to be circumscribed about the pyramid.

Proposition 266 Theorem

If a series of prisms is inscribed in or circumscribed about a triangular pyramid, the sum of the volumes of the prisms approaches the volume of the pyramid as its limit as the number of prisms is increased indefinitely.

Hypothesis. Let V denote the volume and h the altitude of a triangular pyramid. Let v denote the sum of the volumes of a series of inscribed prisms, and v' the sum of a series of circumscribed prisms, all the prisms having equal altitudes.

Conclusion. As the number of prisms is increased indefinitely, v=V and v' V.

Proof. Each inscribed prism is equivalent to the circumscribed prism directly above it.

(?)

Hence, denoting the volume of the lowest circumscribed prism by w,

v' — v = w.

If, now, the number of parts into which h is divided is increased indefinitely, w can be made so small as to become and remain less than any assigned quantity, however small. Hence V v and v' V can be made so small as to become and remain less than any assigned quantity, however small.

(?)

..v V and v' V.

Q. E. D.

Proposition 267 Theorem

Triangular pyramids having equivalent bases and equal altitudes are equivalent.

Let

Hypothesis. A-BCD and A'-B'C'D' are two triangular pyramids having equivalent bases and equal altitudes. their volumes be denoted by V and V' respectively.

Conclusion. V = V'.

Proof. Place the pyramids so that their bases shall lie in the same plane, and let BE be the common altitude.

Divide BE into any number of equal parts.

Through the points of division pass planes to the plane of the bases.

The corresponding sections thus formed are equivalent. (?) Using these sections as upper bases, construct a series of inscribed prisms in each pyramid.

The corresponding prisms are equivalent.

(?)

.. the sum of the volumes of the prisms inscribed in. A-BCD is equivalent to the sum of the volumes of the prisms inscribed in A'-B'C'D'.

Denoting the total volumes by W and W' respectively,

W = W'.

If, now, the number of parts into which BE is divided is increased indefinitely, WV and W'

V.

To be completed by the student, using the method of limits.]

(?)