Chap. VIII.

GEOMETRY.

87.-5

Fig. IMG.

multiply it by the radius, and half the product will be the area: or, multiply the squareof the diameter by *7854, and the product is the area.

To find the Area of a Circle when neither the Diameter northe Circumference are given. When the arc ECF is given, tofind the area of the circle A BCD: draw the chord orright line E F, and divide it into two equal parts, fromwhence draw a perpendicular G C, and measure F G ; thenmultiply EC by itself, and divide the product by G C; thequotient will give E G, to which add G C ; the sum will bethe diameter A C: then by proportion proceed as beforedescribed.

1 'o measure a Circular Ring, as A B C D —First ascertainthe area of the whole circle, and then of the part containedin the lesser one, and deduct one from the other; theremainder will be the area of the ring.

The diameter of the interior circle we will suppose 9 feet;its circumference then will be found to be 28 feet 3 inches

5 parts ; and if we consider the breadth of the circular ringto be 3 feet, we shall have 15 feet for the diameter of theouter circle, the circumference of which will be 47 feet1 inch 8 parts; the arc of this circle will be found equal to176 feet 3 inches: if we then deduct from it the area ofthe inner circle, we shall obtain superficial content of thecircular ring.

The areas or spaces of circles arc to each other as thesquares of their diameters or of their radii, for, as beforeobserved, similar polygons inscribed in circles arc to eachother as the squares of the diameters of the circles; for,conceiving the number of sides of the polygons to be in-creased more and more, or the length of the sides to become lessand less, the polygon approaches nearer and nearer to the circle,till in the end it coincides and becomes in effect equal; hencethe areas of circles, which are the same as polygons, must beto each other as the squares of the diameters of the circles.

To find the Area of a Spiral. — Add together the exterior andthe interior circumference, and multiply their sum by the breadthof the band, and half the product will be the area. To measuresuch a spiral as A, whose breadth BII is 3 feet, its exterior cir-cumference BCD 25 feet 1 inch 8 parts, and its interior 15 feet8 inches 7 parts; add these together, and multiply their sum, 40feet 10 inches 3 parts, by 3 feet, the product of which divided willgive the area of the band.

To find the Area of Sectors. —Multiply the arc C D E by I» C,the radius of the circle; take half the product, which will be thesuperficial content of the section. By the same rule the area ofthe sector X may be ascertained; for example, to measure thefigure A, we multiply the arc CDE, 15ft. 8 in- 6p., by D C 7ft.

6 in. which is the radius of the circle. Half the product will be thecontent of the segment; and if, on adding the area of the sectorX, we find the sum equal the content of the whole circle, weare sure that the calculation is correct.

The sector of a circle is a portion of the area of the circlebounded by two radii and the intercepted arc, and sectors aresaid to be similar when the sides or radii include equal angles.

The area of a sector, then, must be equal to that of a trianglewhose base is equal to the length of the contained arc, andaltitude equal to the radius of the circle.

To find the Area of a Segment _Form on the arc CDE a

sector B E D C, and find the area by the preceding rules; thenfind the area of the triangle, and subtract it from the area of thewhole; the remainder will be the area of the segment. Forexample, the line C E, 12 feet 5 inches 8 parts, and its arc C D E,

12 feet 8 inches 6 parts, to ascertain its area: form on the arc Fig. 1140.

CDE the sector BE DC, and find its superficies; then that of the triangle BEC, andsubtract it from the area of the whole sector, when the remainder will be that ofthe segment A. Should the segment be small, add half the length of the chord to 5 theheight of the arc, and multiply the sum by the whole height of the arc.

Fig. 1147.

Fig. 1148.