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.

Example.

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 ?

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 eafily obtained by having the Side only given. For, as all fimilar or like Solids (efpecially these regular ones) are to one another as the Cube of their like Sides; and their Superficies alfo being fimilar and alike, are therefore to each other as the Squares of their like Sides, we have this

Rule.

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.

Example.

What is the Solidity and Superficies of each of the Regular Solids, fuppofing each Side of them to be 1 Inch, or i Foot, &c?

For the Superficial Contents.

Operation.

Square of 6. Superf. Cont. Squ. of 1. Superf. Cont.

1.732 of the Tetraëdron.

Octaëdron.

Note. By thefe laft Numbers, the Solidity and Superficies of any of the Regular Solids may be found easier than by the former Operations. For, here you need only multiply the Cube of the given Side by the Number expreffing the Solid Content in the Table, to know its Solidity; and the Square of the Side multiplied by the Number expreffing 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 infcribed in a Sphere of any given Dimenfions; we fhall here annex an eafy Geometrical Problem for that Purpose.

Problem 6.

To find the Length of the Sides of the five Regular Solids infcribed in a Sphere of any given Dimenfions.

Example.

Suppofe the Diameter of the given Sphere be 2 Inches, what are the Lengths of the Sides of each of the five Regular Bodies that can be circumfcribed by it?

Conftruction. Let DR be the Diameter of the given Sphere of two Inches; and let Daabb R be (one third of that Diameter). Erect the Perpendiculars ae and ef, and draw the Chords De, Df, e R, and ƒ R. Then will

(ft.) Re be the Length of the Side of the Tetraëdron = 1.62 Inches.

(zd.) De, the Side of the Hexaëdron, 1.15 Inches.

(3d.) Dƒ =ƒR, the Side of the Octaëdron,

Inches.

1.41

(4th.) Cut the Chord De in extreme and mean Proportion, by Problem 23, in h, and D h will be the Side of the Dodecaedron = 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 c G, 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 infcribed in it, as fuppofe the Icofaëdron 1.05 Inches fo is the Diameter of any other Sphere, fuppofe 12 Inches, to 6.3 Inches, the Side of the Icofaëdron infcribed in that Sphere. *

This Problem may be useful to those who want to cut out any of the above Bodies in Wood or Stone to a determinate Size, either for Dials, or Ornaments for Gateways, &c.

* 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 Tetraedron 9.7; Hexaëdron 6.9; O&taëdron 8.4; Dodecaedron 4.2; Icofaëdron 6.3 Inches,