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PRISMS AND PARALLELOPIPEDS.

589. DEF. A prism is a polyhedron of which two faces are equal polygons in parallel planes, and the other faces are parallelograms.

The equal polygons are called the bases of the prism, the parallelograms, the lateral faces, and the intersections of the lateral faces, the lateral edges of the prism. The sum of the areas of the lateral faces of a prism is called its lateral area.

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Right Prism.

Prism.

590. DEF. The altitude of a prism is the perpendicular distance between the planes of its bases.

591. DEF. A right prism is a prism whose lateral edges are perpendicular to its bases.

592. DEF. A regular prism is a right prism whose bases are regular polygons.

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596. DEF. A parallelopiped is a prism whose bases are parallelograms.

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597. DEF. A right parallelopiped is a parallelopiped whose lateral edges are perpendicular to the bases.

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604. DEF. A truncated prism is the part of a prism included between the base and a section made

by a plane oblique to the base.

PROPOSITION I. THEOREM.

605. The sections of a prism made by parallel planes cutting all the lateral edges are equal polygons.

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Let the prism AD be intersected by parallel planes cutting all the lateral edges, making the sections GK, G'K'.

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Proof. The sides GH, HI, IK, etc., are parallel, respectively, to the sides G'H', H'I', I'K', etc.

§ 528 The sides GH, HI, IK, etc., are equal, respectively, to G'H', H'I', I'K', etc.

§ 180 The GHI, HIK, etc., are equal, respectively, to ▲ G'H'I', H'I'K', etc.

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§ 534

§ 203

Q. E. D.

606. COR. Every section of a prism made by a plane parallel to the base is equal to the base; and all right sections of a prism are equal.

Ex. 630. The diagonals of a parallelopiped bisect one another.

Ex. 631. The lateral faces of a right prism are rectangles.

Ex. 632. Every section of a prism made by a plane parallel to the lateral edges is a parallelogram.

PROPOSITION II. THEOREM.

607. The lateral area of a prism is equal to the product of a lateral edge by the perimeter of the right section.

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Let GHIKL be a right section of the prism AD', S its lateral area,

E a lateral edge, and P the perimeter of the right section.

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Therefore, S, the sum of these parallelograms, is equal to

But

Therefore,

E(GH+HI+IK + etc.).

GH+HI+IK + etc. = = P.

S EXP.

Q. E.D.

608. COR. The lateral area of a right prism is equal to the product of the altitude by the perimeter of the base.

Ex. 633. Find the lateral area of a right prism, if its altitude is 18 inches and the perimeter of its base 29 inches.

PROPOSITION III. THEOREM.

609. Two prisms are equal if three faces including a trihedral angle of the one are respectively equal to three faces including a trihedral angle of the other, and are similarly placed.

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In the prisms AI and A'I', let the faces AD, AG, AJ be respectively equal to A'D', A'G', A'J', and similarly placed.

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Therefore, the trihedral angles A and A' are equal.

§ 582

Apply the trihedral angle A to its equal A'.

Then the face AD coincides with A'D', AG with A'G', and

AJ with A'J'; and C falls at C', and D at D'.

The lateral edges of the prisms are parallel.
Therefore, CH falls along C'H', and DI along D'I'.
Since the points F, G, and J coincide with F', G', and J',

§ 589

§ 105

each to each, the planes of the upper bases coincide.

$496

Hence, H coincides with H', and I with I'.

Therefore, the prisms coincide and are equal.

Q. E. D.

610. COR. 1. Two truncated prisms are equal under the hypothesis given in § 609.

611. COR. 2. Two right prisms having equal bases and equal altitudes are equal.

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