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1. The linear side of the icosaedron ABCDEF is 3, what is the solidity ?
AB 7+3 75
-=& AB'y 3 24
2 solidity of the icosaedron. Q. E. D.
If L be put for the linear side, then will 5L'/3=surface of the icosaedron.
Note.—The superficies and solidity of any of the five regular bodies may be found as follows.
RULE 1. Multiply the tabular area by the square of the linear edge, and the product will be the superficies.
2. Multiply the tabular solidity by the cube of the linear edge, and the product will be the solidity.
Surfaces and Solidities of the Regular Bodies.
No. of sides.
4 Tetraedron 1.73205
8 Octaedron 3.46410
0.11785 1.00000 0.47140 7.66312 2.18169
6.70818+7 5x9 13.70818 45
= V 6.85409 x 22.5=2.61803 x 22.5=58.9056=solidity required.
2. Required the solidity of an icosaedron, whose linear side is 1 ?
To find the convex superficies of a cylindric ring.
To the thickness of the ring add the inner diameter, and this sum being multiplied by the thickness, and the product again by 9.8696, will give the superficies required.
1. The thickness of Ac of a cylindric ring is 3 inches, and the inner diameter cd 12 inches; what is the convex superficies ?
* A solid ring of this kind is only a bent cylinder, and therefore the rules for obtaining its superficies, or solidity, are the same as those already given. For let Ac be any section of the solid perpendicular to its axis on, and then Acx 3.14159, &c.=circumference of that section, and Ac i cd (0%) X 3.14159, &c. length of the axis on.
12 + 3x3 x 9.8696 =15* 3 * 9.8696=45x9.8696= 444.132=superficies required.
2. The thickness of a cylindric ring is 4 inches, and the inner diameter 18; what is the convex superficies ?
inches. 3. The thickness of a cylindric ring is 2 inches, and the inner diameter 18; what is the convex superficies ?
To find the solidity of a cylindric ring.
To the thickness of the ring add the inner diameter, and this sum being multiplied by the square of half the thickness, and the product again by 9.8696, will give the solidity.
Whence Ac * 3.14159, &c. X Ac+cd x 3.14159, &c.= Ac + cdx Ac x 3.14159, &c. =Ac+cd x Ac x 9.8696, &c. =superficies required.
3.14159 * Demon. Ac* x.78539, &c.= AO® X
&c. = 1
4 Acé x 3.14159, &c.= area of the section Ac; and AC + cd (on) x 3.14159, &c.=length of the axis on.
Therefore Ac+cd x Ac X3.14159, &c. l=Ac+ cd x I Ac x9.8696. Q.E.D.
This figure being only a cylinder bent round into a ring, its surface and solidity may also be found as in the cylinder, namely, by multiplying the axis or length of the cylinder by the circumference of the ring, or section, for the surface, and by the area of a section for the solidity.
Thus, if c=circumference of the ring, or section, a=area of that section, and l=length of the axis; then will cl=surface of the ring, and al=to its solidity. Which rules are the same as for the cylinder, and may be easily converted into those given in the text.
These rules are indeed so obvious, as to render any demonstration of them altogether unnecessary.
EXAMPLES. 1. What is the solidity of an anchor ring, whose inner diameter is 8 inches, and thickness in metal 3 inches?
8+3 x 3 x 9.8696=11 x 1.5% x 9.8696=11 x 2.25 x 9.8696=24.75 x 9.8696=244.2726=solidity required
:2. The inner diameter of a cylindric ring is 18 inches, and its thickness 4 inches; what is the solidity ?
Ans. 868.5248. 3. Required the solidity of a cylindric ring, whose thickness is 2 inches, and its inner diameter 12.
Ans. 138.1744. 4. What is the solidity of a cylindric ring, whose thickness is 4 inches, and inner diameter 16 ? Ans. 789.568.