Εικόνες σελίδας
PDF
Ηλεκτρ. έκδοση

Side, the Square Root of the Remainder would be the Height of the Pyramid, as before.

Square of 12 is 144

Square of 6.9283 is 48

96 Remainders; whofe

Root being extracted gives the perpendicular Height of the Pyramid the fame as before.

Of the Octaedron.

The Octaedron is a Solid contained under eight equal and equilateral Triangles, which may be conceiv'd to confift of two quadrangular Pyramids, of equal Bafes, joined together: Therefore, to meafure the Solidity of this Body, it will be fufficient to confider it as two fquare Pyramids, the Sides of whofe Bafes are equal to the given Side of the Triangles, under which it is contained.

The Side of an Octaedron being 12, to find the folid and fuperficial Content.

The Sides of the Bafes being 12, their Area will be 144, by Art. 52. If the Square of Half the given Side be fubtracted from the Square of the perpendicular Height of the Triangle, the Square Root of the Remainder is the Height of each Pyramid.

The Side of the Triangle being the fame here as in the Tetraedron, the Square of the Perpendicular of the Triangle will be 108, as before; from which fubtracting 36, the Square of Half the given Side, the Remainder is 72; whose square Root multiply'd by 48, the third Part of the Area of the Base, gives the folid Content of one Pyramid; which doubled gives the Solidity of the Octaedron.

72.0000 (8.485 Height of each 64 (Pyramid. 164 (800

[blocks in formation]

8.485

48

67880

33940

407.28

2

814.56 Solidity.

The

The Area of the Bafe being the fame here as in the Tetraedron, the Superficies will be the Double of the Superficies of that, as here are Eight Triangles of the fame Dimenfions.

Of the Dodecaedron.

The Dodecaedron is a Solid contain'd under twelve equilateral Pentagons, and may be conceiv'd to confift of twelve pentagonal Pyramids, of equal Bafes and Altitude, whose Vertexes meet in the Center of the Dodecaedron: Therefore, if we find the folid Content of one of thofe Pyramids, that multiplied by 12 gives the Solidity of the Dodecaedron.

The Side of the Dodecaedron being 12, to find the folid and fuperficial Content,

To find the Solidity of a Pyramid, we must firft find the perpendicular Height: Now, the Side of a Dodecaedron being given, the flant Height of each Pyramid of which it confifts, meafur'd from the Angles to the Vertex, is to the given Side of the Pentagon, as I is to .7136444; and therefore this will be to that, as .7136 to 1, nearly.

[blocks in formation]

Next find the Diftance of the Center of the pentagonal Bases from the Angles: In all regular Pentagons, as 117556 is to 100000, fo is the Side of the Pentagon to the Distance of the Center from the Angles.

As

As 117556 is to 100000, fo is 12.

12

117556) 1200000 (10.2079
117556

244400

235112

928800

822892

1059080

1058004

1076

If the Square of the Distance from the Center be fubtracted from the Square of the flant Height, the Square Root of the Remainder is the perpendicular Height of the Pyramid.

16.816

16.816

100896
16816

134528

100896
16816

282.777856

104.201222

[blocks in formation]
[blocks in formation]

The Square of 12, the Side of the Pentagon, multiplied into the tabular Number for Pentagons, in the Table of Art. 60, will give the Area of the Base.

[blocks in formation]

13229-7799392 Solidity of the Dodecaedron.

If the laft Extraction had been farther continued, the perpendicular Height would be 13.3632, whofe third Part would be 4.4544; in which Cafe the Solidity would be 13241, and fomewhat more.

For the Superficies, the Area of the Pentagon multiplied by 12 gives the Surface of the whole Dodecaedron.

247.748688

12

2972.984256 Superficies.

Of the Eicofiedron.

The Eicofiedron is a Solid contained under twenty equal and equilateral Triangles, and may be conceived to confift of twenty equal triangular Pyramids, whofe Vertexes all meet in the Center: Hence, if we find the folid Content of one of those Pyramids, that being multiplied by 20 gives the Content of the Eicofiedron.

The Side of an Eicofiedron being 12, to find the folid and fu perficial Content: We must first find the Distance of the Center of the Bafes from the Angles, and then the flant Height

the Pyramids; from which the perpendicular Height may

be

be found, by fubtracting the Square of the Distance of the Center from the Square of the flant Height, and extracting the Square Root of the Remainder.

The Distance of the Center of the Triangle from the Angles we have found in the Tetraedron (the Triangles being here of the fame Dimensions) to be 6.9283; and for the flant Height of the triangular Pyramids of the Eicofiedron, this is the Proportion, as 1.05146 is to one, fo is the given Side of the Eicofiedron to the flant Height of the Pyramids, of which it confifts.

As 1.05146 is to 1, fo is 12.

[blocks in formation]

2458

The Square of the Diftance of the Center from the Angles being fubtracted from the Square of the flant Height, the Square Root of the Remainder is the Height of the Pyramid.

Square of 11.427 is 130.2497

Square of 6.9283 is 48

[ocr errors][merged small][merged small][merged small][merged small]
[blocks in formation]
« ΠροηγούμενηΣυνέχεια »