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556. DEF. A Regular pyramid is a pyramid whose base is a regular polygon, and whose vertex is in the perpendicular to the base at its centre.

557. DEF. The Axis of a regular pyramid is the straight line joining its vertex with the centre of the base.

558. DEF. The Slant height of a regular pyramid is the altitude of any lateral face.

559. DEF. A pyramid is triangular, quadrangular, pentangular, etc. according as its base is a triangle, quadrilateral, pentagon, etc. A triangular pyramid formed by four faces (all of which are triangles) is a tetrahedron.

560. DEF. A Truncated pyramid is the portion of a pyramid included between its base and a section cutting all its lateral edges.

561. DEF. A Frustum of a pyramid is a truncated pyramid in which the cutting section is parallel to the base.

562. DEF. The base of the pyramid

is called the Lower base of the frustum, and the parallel section, its Upper base.

563. DEF. The Altitude of a frustum is the perpendicular distance between the planes of its bases.

564. DEF. The lateral faces of a frustum of a regular pyramid are the trapezoids included between its bases; the lateral surface is the sum of the lateral faces; the Slant height of a frustum of a regular pyramid is the altitude of any lateral face.

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565. If a pyramid be cut by a plane parallel to its base, I. The edges and altitude are divided proportionally; II. The section is a polygon similar to the base.

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Let the pyramid V-A B C DE, whose altitude is VO, be cut by a plane a b c d e parallel to its base, intersecting the lateral edges in the points a, b, c, d, e, and the altitude in o.

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II. The section abcde similar to the base A B C D E. I. Suppose a plane passed through the vertex Vll to the base. Then the edges and the altitude will be intersected by three

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(if straight lines be intersected by three || planes, their corresponding segments

are proportional).

II. The sides a b, bc etc. are parallel respectively to A B, BC,

etc.,

(the intersections of || planes by a third plane are || lines) ; ..abc, bed etc. are equal respectively to

BCD etc.,

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ABC,

§ 462

(if two not in the same plane have their sides respectively || and lying in

the same direction, they are equal).

..the two polygons are mutually equiangular.

Also, since the sides of the section are to the corresponding sides of the base,

A Vab, Vbc etc. are similar respectively to ▲ VAB,

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.. the polygons have their homologous sides proportional; ... section a b c d e is similar to the base A B C D E. § 278

Q. E. D.

566. COROLLARY 1. its base is to the base as tex is to the square of the altitude of the pyramid.

Any section of a pyramid, parallel to the square of its distance from the ver

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(similar polygons are to each other as the squares of their homologous sides).

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567. COR. 2. If two pyramids having equal altitudes be cut by planes parallel to their bases, and at equal distances from their vertices, the sections will have the same ratio as their bases.

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=

=

Now, since Vo V'o', and VO V'0',

abcde: ABCDE:: abc: A'B'C'.

Whence abcde: a'b'c:: ABCDE: A'B'C'. § 262 568. COR. 3. 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 XV. THEOREM.

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

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Let V-ABCDE be a regular pyramid, and V H its

slant height.

We are to prove the sum of the faces V AB, VBC, etc. (AB+ BC, etc.) X VH.

=

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(oblique lines drawn from any point in a 1 to a plane at equal distances

from the foot of the are equal).

..A VAB, VB C, etc. are equal isosceles A,

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whose bases are the sides of the regular polygon and whose common altitude is the slant height V H.

Now the area of one of these A, as VA B,= 1⁄2 base AB X altitude VH,

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.. the sum of the areas of these ▲, that is, the lateral area of the pyramid, is equal to the sum of their bases

(AB+ BC + CD, etc.) X V H.

Q. E. D.

570. COROLLARY 1. The lateral area of the frustum of a regular pyramid, being composed of trapezoids which have for their common altitude the slant height of the frustum, is equal to one-half the sum of the perimeters of the bases multiplied by the slant height of the frustum.

571. COR. 2. The dihedral angles formed by the intersections of the lateral faces of a regular pyramid are all equal. § 492

PROPOSITION XVI. THEOREM.

572. Two triangular pyramids having equivalent bases and equal altitudes are equivalent.

X

S'

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Let S-ABC and S'-A'B'C' be two triangular pyramids having equivalent bases A B C and A'B'C' situated in the same plane, and a common altitude A X. We are to prove S-ABC S'-A' B' C'.

Divide the altitude A X into a number of equal parts,

and through the points of division pass planes I to the planes of their bases, intersecting the two pyramids.

In the pyramids S-A BC and S'-A'B'C' inscribe prisms whose upper bases are the sections DEF, G H I, etc., D' E' F', G'H'I', etc.

The corresponding sections are equivalent,

§ 568 (if two pyramids have equal altitudes and equivalent bases, sections made by planes to their bases and at equal distances from their vertices are equivalent).

.. the corresponding prisms are equivalent, $ 544 (prisms having equivalent bases and equal altitudes are equivalent).

Denote the sum of the prisms inscribed in the pyramid S-A BC, and the sum of the corresponding prisms inscribed in the pyramid S-A' B' C' by V and V' respectively.

Then

V = V'.

Now let the number of equal parts into which the altitude A X is divided be indefinitely increased;

The volumes V and V' are always equal, and approach to the pyramids S-A B C and S'-A' B'C' respectively as their limits.

Hence

S-ABC S'-A' B' C'.

§ 199

Q. E. D.

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