バラ

PYRAMIDS

608. A pyramid is a polyhedron, one of whose faces is a polygon and whose other faces are all triangles having a common vertex.

The lateral faces of a pyramid are the triangles.

The lateral edges of a pyramid are the intersections of the lateral faces. The vertex of a pyramid is the common vertex of all the lateral faces.

The base of a pyramid is the face opposite the vertex.

The lateral area of a pyramid is the sum of the areas of the lateral faces. The total area of a pyramid is the sum of the lateral area and the area of the base.

The altitude of a pyramid is the perpendicular distance

from the vertex to the plane of the base.

A triangular pyramid is a pyramid whose base is a tri

angle. It is called also a tetrahedron. (See 566.)

609. A regular pyramid is a pyramid whose base is a regular polygon and whose altitude, from the vertex, meets the base at its center.

The slant height of a regular pyramid is the line drawn in a lateral face, from the vertex perpendicular to the base of the triangular face. It is the altitude of any lateral face.

610. The frustum of a pyramid is the part of a pyramid included between the base and a plane parallel to the base.

The altitude of a frustum of a pyramid is the perpendicular distance between the planes of its bases.

The slant height of the frustum of a regular pyramid is the perpendicular distance, in a face, between the bases of that face.

A truncated pyramid is the part of a pyramid included between the base and a plane cutting all the lateral edges.

PRELIMINARY THEOREMS

611. THEOREM. The lateral edges of a regular pyramid are all equal. (See 520, II.)

612. COR. The lateral faces of a regular pyramid are equal isosceles triangles.

613. COR. The lateral edges of the frustum of a regular pyramid are all equal. (Ax. 2.)

614. THEOREM. The lateral faces of the frustum of a regular pyramid are equal isosceles trapezoids. (See 500.)

615. THEOREM.

The lateral faces of the frustum of any pyramid are trapezoids. (?).

616. THEOREM. The slant height of a regular pyramid is the same length in all the lateral faces.

Ex. 1. Prove that the bases of any frustum of a pyramid are inutually equiangular.

Ex. 2. The foot of the altitude of a regular pyramid drawn from the vertex, coincides with the center of the circles inscribed in, and circumscribed about, the base.

Ex. 3. The sum of the medians of the lateral faces of the frustum of a trapezoid is equal to half the sum of the perimeters of the bases.

Lateral area AB · 8+ BC. 8+ etc. (Ax. 2).

=

Adding, lateral area AI= {(P+p) · 8. (Explain.) Q.E.D.

Ex. 1. The slant height of a regular pyramid whose base is a square, of which each side is 8 ft., is 15 ft. Find the lateral area; the total area. Ex. 2. A regular pyramid stands on a hexagonal base 16 in. on a side, and the slant height is 2 ft. Find the lateral and total areas.

619. THEOREM. If a pyramid is cut by a plane parallel to the base:

I. The lateral edges and altitude are divided proportionally.

II. The section is a polygon similar to the base.

Given: Pyr. O-ABCDE; plane FI to the base; altitude = OL.

Proof: I. Imagine a plane through o || to plane AD.

This plane is to oz (512), and || to FI (?) (505).

II. FG is to AB, GH is || to BC, etc. (?) (500).

../FGH =

LABC; /GHI

=

▲ BCD; etc. (?) (515).

That is, the polygons are mutually equiangular. Also, ▲ OFG is similar to ▲ OAB; ▲ OGH to ▲ OBC; etc. (?) (316).

Q.E.D.

.. section FI is similar to base AD (?) (312).

Ex. 1. The bases of the frustum of a regular pyramid are equilateral triangles whose sides are 12 in. and 20 in., respectively. The slant height is 40 in. Find the lateral area; the total area.

Ex. 2. The bases of frustum of a regular pyramid are regular hexagons whose sides are 8 and 18, respectively. The slant height is 25. Find the lateral area and total area.

620, THEOREM.

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

O'L';

Given: Pyramids 0-ABCDE and o'-PQRS; alt. OL base AD base PR; sections FI and TV || to bases; OM = O'M'.

Proof: FG is to AB; TU is | to PQ, etc. (?) (500).

AOFG and OAB are similar, also A O'TU and O'PQ (?) (316).

Now section FI is similar to base AD, and section TV is

similar to base PR (?) (619, II).