Of the Oilaëdron. Def. An Oftaëdron is a Solid composed of Eight equa Pyramids, whose Tops all meet in a Point at the Cente of the Solid; the Base of each being an equilateral Triangle and each equal to the other ; or, it is composed of tw Quadrangular Pyramids joined together by their Base The Superficies, therefore, is equal to eight Times th Area of one Triangle, and the Solidity equal to the Soli dity of the eight compofing Pyramids, or to two Qua drangular ones. A fimilar Figure to this here delineated, being draw upon Pafteboard, cut half through in the Lines, folde up, and glued together, will give the Learner an adequate Idea of an Ostaëdron, as composed either of eight Equila. teral Triangular Pyramids, or of two Quadrangular ones.. Problem 3. Rule. Multiply the Area of the Square Base in the Middle by Part of the Height of both Pyramids, and the Product will be the Solid Content required. Crample. Suppose ABCD be an Oftaëdron, whose Side AB= BC=CD, &c. is 6 Inches ; the Height of one of its Sides, as E F, 5.196 Inches; and the Perpendicular Height of the two quadrangular Pyramids BD 8.484 Inches; what is its Solid Content? B E Operation. 6, the Side, multiplied by 6, is 36, the Area of the Square in the Middle, which multiplied by 2.828, one-third of both Pyramids, gives 101.808 Inches, the Solid Content fought. For the Superficial Content, multiply 5.196, the Height of one Triangle, by 3, half the Side of the Base, and that Product multiply by 8, the Number of Triangles, gives 124.704 Inches, the Superficial Content. Of the Dodecaëdron. Def. A Dodecaëdron is a Solid composed of twelve equal Pyramids, whose Tops all meet in a Point at the Center of the Solid; the Base of each Pyramid being an equilateral Pentagon, and equal to each other. The Superficies of such a Body is therefore equal to twelve Times the Area of one Pentagon; and the Solidity is equal to the Solidities of the twelve composing Pyramids. If there be drawn upon Pasteboard a Figure like the following Projection; and the Lines be cut half through, folded up and glued together, the several Pentagons will then form the true Figure of a Dodecaëdron. Problein 4. Rule. Find the Solid Content of one of the Pyramids, and multiply it by 12 (the Number of Pyramids contained in the Figure), the Product will be the Solid Content. Érample. Let ABCDE, &c. represent a Dodecaëdron, each Side of which is equal to 6 Inches; the Height of one of the Pentagons, as OP, is 4.129 Inches; and the Altitude of the whole Figure 13.362 Inches; half of which is the Altitude of one of the Pyramids, viz. 6.681 Inches; what is its Solid Content? F Operation. 4.129, Height of one Pentagon, multiplied by 15, half the Sum of its 5 Sides, gives 61.9350 for the Area of one of the Pentagons; which multiplied by 2.227, onethird of the Height of each Pyramid, gives 137.929245, the Content of one Pyramid ; which multiplied again by 12, the Number of Pyramids, gives 1655.150940 Inches, the Solidity required. For the Superficial Content, multiply 61.935, the Area of one Pentagon, by 12, the Number of Pentagons, and the Product gives 743.220 Inches, for the Superficial Content. Of the Icofaëdron. Def. An Icofaëdron is a Solid made up of twenty Pyramids, whose Tops all meet in a Point at the Center of the Body; the Base of each Pyramid being an Equilateral Triangle, and equal to each other. The Superficies, therefore, is equal to twenty Times the Area of one Triangle; and the Solidity equal to the Solidities of the twenty composing Pyramids. We have added the annexed Figure, that the Learner, by drawing a fimilar one upon Pasteboard, cutting the Lines half through, folding them up together, as directed before, for the other Regular Solids, may conceive a perfect Idea of the Figure and Dimensions of an Icofaëdron. |