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radius of its inscribed circle) by the length of one side, and this product again by the number of sides; and half the product will be the area of the polygon.
[For a table of the areas of regular polygons, see pages 67, 68.]
To find the Area of a Trapezium, fig. 8. RULE 1.-Draw a diagonal line, a c, to divide the trapezium into two triangles; find the areas of these triangles separately, and add them together.
RULE 2.-Divide the trapezium into two triangles, by the diagonal a c, and let two perpendiculars, b ƒ, and de, fall on the diagonal from the opposite angles; then, the sum of these perpendiculars multiplied by the diagonal, and divided by 2, will be the area of the trapezium.
To find the Area of a Trapezoid, fig. 9.
RULE 1.-Multiply the sum of the two parallel sides, a h, d c, by a p, the perpendicular distance
between them, and half the product will be the
RULE 2.-Draw a diagonal, a c, to divide the trapezoid into two triangles; find the areas of those triangles separately, and add them together.
To find the Area of an Irregular Polygon, abcdefg, fig. 10.
RULE.-Draw diagonals to divide the figure into trapeziums and triangles; find the area of each separately, by either of the rules before given for that purpose; and the sum of the whole will be the area of the figure.
To find the Area of a Long Irregular Figure, dcab, fig. 11.
RULE.-Take the breadth in several places, and at equal distances from each other; add all the breadths together, and divide the sum by this number, for the mean breadth; then multiply the mean
breadth by the length of the figure, and the product
will be the area.
To find the Circumference of a Circle when the Diameter is given; or the Diameter when the Circumference is given.
RULE 1.-Multiply the diameter by 3·1416, and the product will be the circumference; or divide the circumference by 3.1416, and the quotient will be the diameter.
RULE 2.—As 7 is to 22, so is the diameter to the circumference. As 22 is to 7, so is the circumference to the diameter.
RULE 3-AS 113 is to 355, so is the diameter to the circumference. As 355 is to 113, so is the circumference to the diameter.
To find the Area of a Circle.
RULE 1.-Multiply the square of the diameter by 7854; or the square of the circumference by
07958; the product, in either case, will be the
RULE 2.-Multiply the circumference by the diameter, and divide the product by 4.
RULE 3.—As 14 is to 11, so is the square of the diameter to the area. Or, as 88 is to 7, so is the square of the circumference to the area.
To find the length of any Arc of a Circle.
RULE 1.-From 8 times the chord of half the arc, a c, fig. 12, subtract the chord, a b, of the whole arc; one-third of the remainder will be the length of the arc, nearly.
As 180 is to the number of degrees in the arc;
To find the Area of a Sector of a Circle, fig. 13. RULE 1.-Multiply the length of the arc, a d b, by half the length of the radius, a c; the product will be the area.
RULE 2.-AS 360 degrees is to the number of degrees in the arc of the sector; so is the area of the circle to the area of the sector.
To find the Area of a Segment of a Circle, fig. 12. RULE 1.-To the chord, a b, of the whole arc
add the chord, a c, of half the arc and one-third of it more. Then multiply the sum by the versed sine, or height of the segment c d, and four-tenths of the product will be the area of the segment.
RULE 2.-Divide the height, or versed sine, by the diameter of the circle, and find the quotient in the column of versed sines, at the end of Mensuration of Solids.
Then take out the corresponding area in the next column on the right-hand, and multiply it by the square of the diameter, for the answer.
To find the Area of a Circular Zone, fig. 14. RULE 1.-When the Zone is less than a Semicircle, to the area of the trapezoid, a b c d, add the area of the circular segments, a c and b d; the sum is the area of the zone.
RULE 2.- When the Zone is greater than a Semicircle, to the area of the parallelogram, e f g h, add the area of the circular segments, e k g and flh; the sum is the area of the zone.
To find the Area of a Circular Ring, or Space, included between two Concentric Circles.
RULE. Find the areas of the two circles separately; then the difference between them will be the area of the ring.