Chat. VIII.

GEOMETRY.

877

To measure the superficies of the truncated cone

To find the Superficies of Cones , it is only necessary to mul-tiply the length of the side by the circumference, and take halfthe product. Thus in the cone A, whose side B E is 24 feet,and the circumference of the base BCD 19 feet, 24 x 19 =

l 5 ? = 228.

2

A, whose diameter at the base B E is 8 feet, and at CG 6 feet:add together these diameters, and take the half for a mean,which is 7 feet; this multiplied by 8, the length of the side BC,gives the product 56, from which the square root is to be ex-tracted.

It must be borne in mind that k

with a radius of 7 feet 6 inches,which is equal to H I, we candescribe a circle FIKLM, whichshall contain an area equal to thesuperficies of the cone A.

To find the Area of a Cylinder. —

Multiply the height by the circum-ference ; the product will be the _

area, to which add that of the two Fig. 1155. Fig- 1156.

^To measure the cylinder A, multiply the height B C, which is 21 feet, by the circum-ference CDE, which is 44 feet, and the product, 924 feet, will be the superficies ol thecylinder, without comprising the area of the two ends.

If the cylinder be irregular, as that at F, multiplythe less height, IK, 18 feet, by the circumference,

GUI, 44 feet, and the product will be 572. Sub-tract the shortest height, IK, 13 feet, from thelongest, GL, 21 ; multiply the remainder, 8, by thecircumference, and take half the product, 176, andadd it to 572, which will be 748, the superficies ofthe cylinder.

Mixed or irregular Figures , as ovals or parabolas,must be divided into small segments, triangles, and

squares before their area can be ascertained. The *" h

oval shown at A for instance may be divided into Fig. 1157. Fig. 1158.

two segments, B C F h

and BCG , and the areafound by the precedingrules.

Figures bounded byseveral curved linesmust be enclosed with-in a triangle, as that atFDBG, by the linesI and K : the area ofthe equilateral trianglebeing found, the seg-ments may be after-wards found, and theirarea deducted: whenthe form is very much

varied, as that com- Fig. 1159. Fig. 1160.

prised within the irregular line HIKLMN, it must be similarlytreated, dividing it into triangles, segments of circles, or anyother regular figures.

On fiat and round Superficies "We at once see in the diagrams

G and H the effect of houses so placed that they radiate froma centre, and are drawn perpendicular to the base line. Onhilly or mountainous land no more grass or trees can grow thanwould do on its base if the whole were levelled: trees riseperpendicular, or tend to the centre of the earth, and not to thesurface on which they stand ; consequently the superficial con-tent or area of a mountain is that of its base; so it is with a field, 11G2

where no more blades of corn can stand upon uneven ground

than would on the level plane comprised within the same boundary. In carrying a fence

Fig. 1161.