EXAMP. 1. In a triangular pyramid, the height BE, being 48, and each side of the base 13: the base being a triangle, let the perpendicular height DE be 11; to find the content.

5.5

half ED.

EXAMP. 2. In a quadrangular pyramid, the height BE being 48, and each side of the base 13, to find

the content.

B

EXAMP. 3. To measure a Cone--The diameter AC being 13, and the height BD 48, to find the

87

E

D

D

Note. The superficial content of all pyramids is found by taking the sum of the several areas, which compose them. That of a cone, by multiplying the circumference of the base into half the line joining the vertex and any point in that circumference, and adding the area of the base to the product.

ART. 33. To measure the Frustum of a Pyramid.

Definition. The frustum of a pyramid is what remains after the top is cut off by a plane parallel to the base, and is in the form of a log greater at one end than the other, whether round, or hewn three or four square, &c.

RULE. If it be the frustum of a square pyramid, multiply the side of the greater base by the side of the less; to this product add one third of the square of the difference of the sides, and the sum will be the mean area between the bases; but if the base be any other regular figure, multiply this sum by the proper multiplier of its figure in the Table, Art. 11. and the product will be the mean area between the bases: lastly, multiply this by the height, and it will give the height of the frustum.

EXAMP. 1. In the frustum of a square pyramid the

side of the greater base AD=15, the side of the less, B BC=6, and the height EF-40, to find the content.

Or, if it be a tapering square stick of timber, take the girth of it in the middle; square of the girth (or multiply it by itself in inches) then say, as 144 (inches) to that product; so is the length, taken in feet, to the content in feet.

EXAMP. 2. What is the content of a tapering square stick of timber, whose side of the largest end is 12 inches, of the least end, 8, and whose length is thirty feet.

One fourth of the girth in the middle=10, and 10x10=100, the area in the middle; then, as 144: 100 :: 30 feet: 20 83 feet the content.

By the Sliding Rule.

of the circumference on C, and against the

Set 12 on D to length on D is the answer on C.

By Gunter.

The extent from 12 to 1 of the circumference doubled, or twice turned over, will reach from the length to the content. EXAMP. 3. In the frustum of a triangular pyramid,

F

the side of the greater base AC=15, as before, the BD side of the less BD-6, and the height EF=40, to

Or, if it be a tapering three square stick of timber, you may find the area midway from end to end, then, as 144 is to that area, so is the length, taken in feet, to the content in feet.

EXAMP. 4. To measure the Frustum of a Cone.

RULE. Multiply the diameters of the two bases together, and to the product add one third of the square of the difference of the diameters then multiplying this sum by 7854, it will be the mean area between the two bases, which being multiplied by the length of the frustum, will give the solid content.

Or, to the areas of the top and bottom add the square root of the product of those areas, and the sum, multiplied by one third of the height of the frustum, will give the solidity.

When figures run uniformly taper; but not to a point (they being considered as portions of the cone or pyramid) we may find the solidity by supplying what is wanting to complete the figure, and then deducting the part cut off.

A general rule for completing every straight sided solid, whose ends are parallel and similar.

As the difference of the top and bottom diameters is to the perpendicular height, (or depth which is the same :) so is the longest diameter to the altitude of the whole cene or pyramid.

EXAMP. 1. The former cone in Art. 32, Examp. 3, being cut off in the middle, the greater diameter AC is 13, the less BD 61, and height EF 24, to find the content of the frustum.

13
6.5

AC 13 inches.

BD 6.5 inches.

65

65 difference.

EXAMP. 2. What number of barrels, each 32 gallons of Ale measure, is contained in a cistern whose largest diameter is 6 feet, and smallest diameter 5 feet, and whose depth is 8 feet?

6-512

6x5=30

3

301 mean diameter.

**7854

23.5620

•2618

23.9238 mean area.

8

190.5904 content in feet.

1728 inches in a solid foot.

329340 2112 cubic inches, which divided by 9024, the cubic inches in a barrel or 32 gallons, gives 36.5 barrels nearly, Ans.

If the answer had been required in Beer Measure, where the barrel contains 36 gallons, the answer would have been 32-4 barrels.

Note. If when the end diameters of a conical cistern are given, it is required to find the length of the cistern to contain a certain number of barrels; divide the cubic feet contained in the number of barrels by the mean area, and the quotient will be the height.

Let the mean area be as in the last Ex. to find the length of the cistern to contain 50 barrels of 32 gallons of Ale measure.

261.11111 &c. cubic feet in 50 barrels, which divided by 23 8238, the mean area, gives 10.95 feet, for the length of the cistern, Ans.

To find the diameters of the cistern, when the content, and length, and difference of the diameters, are given, see Art. 53.

ART. 34. To measure a Sphere or Globe.

Definition. A sphere or globe is a round solid body, in the middle of which is a point, from which all lines drawn to the surface are equal.

RULE. Multiply the cube of the diameter by 5236, and the product will be the solid content.

Or, multiply the circumferenee by the diameter, which will give the superficial content; then multiply the surface by one sixth of the diameter, and it will give the solidity.

Or, multiply the cube of the diameter by 11, and the product divided by 21, will give the solidity.

EXAMP. The diameter, AB, of a globe, is 4.5 feet; to find the solid content.