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Problem 5. To find the Solidity of an Icofaëdron.
Rule. Find the Content of one of the Triangular Pyramids, which multiply by 20, the Number of all the Pyramids, and the Product will be the Solid Content required.
Crample. Let ABCDE, &c. be an Icofaëdron, each side of which is 6 Inches; the Perpendicular of one of the Triangles, as GK, is 5.196 Inches; the Height of the whole Figure is 9.069 Inches; half of which is 4.534 Inches, the Altitude of one of the Pyramids; what is its Solid Content?
D Operation. Multiply 15.588, the Area of one Triangle, by 1.5115, the third Part of the Height of one Pyramid, the Product 23.561262 will be its Content, which multiplied by 20, the Number of all the Pyramids, gives 471.24612 Inches, the Solidity of the whole Icofaëdron.
For the Superficial Content, multiply 15.588, the Area of one Triangle, by 20, the Number of all the Triangles, and the Product 311.76 is the Superficial Content.
From the Solidity and Superficies thus found of the foregoing Bodies, the Solidity and Superficies of any other like Body may be easily obtained by having the Side only given. For, as all fimilar or like Solids (especially these regular ones) are to one another as the Cube of their like Sides; and their Superficies also being similar and alike, are therefore to each other the Squares of their like Sides, we have this
As the Cube of the Side of any of the foregoing Solids is to its Solid Content, fo is the Cube of the side of any other like Solid to its Solid Content.
And, as the Square of the Side is to its Superficial Content. fo is the Square of the Side of the like Body to its Superficial Content.
What is the Solidity and Superficies of each of the Regular Solids, supposing each Side of them to be 1 Inch, or i Foot, &c?
For the Solid Contents.
Cube of 6.
Solid Content, Cube of 1. Solid Content,
1.0000 of the Hexaëdron. 101.808
.4714 + OEtaëdron. 1655.1509
7.663 + Dodecaëdron. 471.2461
2.1816 - Icofaëdron.
For the Superficial Contents.
Dperation Square of 6. Superf. Cont. Squ. of 1. Superf. Cont. 62.352
1.732 of the Tetraëdron, 216.
6.000 Hexaëdron. 124.704
3.464 Ostaëdron. 743.22
20.645 Dodecaë dron. 311.76
Note. By these last Numbers, the Solidity and Superficies of any of the Regular Solids may be found eafier than by the former Operations. For, here you need only multiply the Cube of the given Side by the Number expressing the Solid Content in the Table, to know its Solidity; and the Square of the Side multiplied by the Number expressing the Superficial Content will give the Superficies of that Body.
As it may be useful to determine the Length of the Sides of any of the Regular Bodies inscribed in a Sphere of any given Dimensions; we shall here annex an easy Geometrical Problem for that Purpose.
Problem 6. To find the Length of the Sides of the five Regular Solids inscribed in a Sphere of any given Dimensions.
Example. Suppose the Diameter of the given Sphere be 2 Inches, what are the Lengths of the Sides of each of the five Rea gular Bodies that can be circumscribed by it?
Construction. Let DR be the Diameter of the given Sphere of two Inches; and let Da= ab = b R be (one third of that Diameter). Erect the Perpendiculars ae and of, and draw the Chords De, Df, e R, and f R. Then wil
(ift.) Rebe the Length of the Side of the Tetraëdron = 1.62 Inches.
(2d.) De, the side of the Hexaëdron, = 1.15 Inches.
(3d.) Df=fR, the side of the Ostaëdron, = 1.41 Inches.
(4th.) Cut the Chord De in extreme and mean Proportion, by Problem 23, in h, and D h will be the Side of the Dodecaëdron = 0.71 Inches.
(5th.) Set up the Diameter DR perpendicularly at R, and from the Center c, to the Top at G, draw the Line cG, cutting the Arch at k, and draw the Chord k R, which will be the side of the Icofaëdron = 1.05 Inches.
If any of these five Bodies were required to be cut out of a Sphere of any other Diameter, the Rule will always hold,
As the Diameter of the Sphere 2 Inches, is to the Side of any one Solid inscribed in it, as suppose the Icofaedron 1.05 Inches : : so is the Diameter of any other Sphere, fuppofe 12 Inches, to 6.3 Inches, the Side of the Icojaëdron inscribed in that Sphere.
This Problem may be useful to thofe who want to cut out any of the above Bodies in Wood or Stone to a determinate Size, either for Dials, or Ornaments for Gate
* In this Manner the Sides of all the Regular Solids inscribed in a Sphere of 12 Inches may be easily found to be as under:
The Tetraëdron 9.7; Hexaëdron 6.9; Oftaëdron 8.4; Dodecaëdron 4.2; Icofaëdren 6.3 Inches,