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Prob. 6. To find the area of a trapezoid.
Multiply the half sum of the parallel sides by the perpendicular distance between them, and the product will be the area.
Ex. 3. What is the area of a board or plank in the form of a trapezoid, being if. 7i. at one end, 2f. 3i. at the other end, and 8f, 6.. long?
Prob. 7. To find the area of any regular polygon. Multiply half the perimeter of the figure by the perpendicular, falling from its centre upon one of the sides, and the product will be the area of the polygon.
Ex. 1. Requireth the area of a regular pentagon ABCD, whose side AB, or BC, &c. is 6 feet, and the perpendicular FG 4 feet.
15 half the perimeter.
60 feet, equal the area required
Er. 2. How many feet of ground does an hexangular building cover, each side of the base being 8. 3i. and the perpendicular 7 feet ?
24 9 half the sum of the sides.
Prob. 8. To find the area of a polygon, when the side only is given.
Multiply the square of the given side of the polygon by that num. ber which stands opposite to its name in the following Table, and tha product will be the area.
In the above Table, those multipliers marked with the sign +, are too small; on the contrary, those marked –, are too great : I have only given this Table to five places of decimals, being exact enough for most practical purposes.
Ex. 1. Requireth the area of a pentagon, each side being 14 feet.
Prob. 9. The diameter of a circle being given, to tind the circumference; or the circumference being given, to find the diameter.
Method 1. As 7 is to 22, so is the diameter to the circumference nearly ; or, as 22 is to 7, so is the circumference to the diameter nearly.
Ex. 1. What is the circumference of a circle, whose diameter is 12 feet?
7 : 22 :: 12
Ex. 2. What is the diameter of a circle, whose circumference is 03 feet ?
22 : 7 :: 63
f. i. ii. 22)441(2006
The most practical method to find the circumference of a circle from its diameter, is the following:
Multiply the diameter by 3 : add í part of the diameter to the product, the sum will be the circumference, the same as in Methol i.
Er. 1. What is the circumference of a circle, whose diameter is df. 6i.?
Er. 2. What is the circumference of a semicircular vault, whose dianieter is 16f. 3in. ?
in. 16 5
2)51 7 1
feet 25 9 6 answer nearly.
Method 2. Multiply the dianeter by 3:1416, and the product will be the circumference.
Or divide the circumference by 3:1416, and the quotient will be the diameter.
This inethod comes nearer to the truth than the foregoing, but for practical purposes, Method I will be sufficiently near.
Prob. io. The chord and height of a segment being given, to find the chord of half the arc.
To the square of the balf chord, add the square of the versed sine, and the square root of the sum will be the length of the chord of half
Lix. The chord AC being 48 feet, and the versed sine DB 18 feet, what is the length of the chord AB or BC of half the arc ? 2)48 18
00 Prob. 11. To find the length of any arc of a circle, the half chord and chord of the whole arc being given.
Subtract the chord of the whole arc from double the chord of the half arc : add one-third of the remainder to the double chord cf the half
arc, and ihe sum will be nearly equal to the length of the arc. Er. 1. What is the length of the arc ABC, of which the chord AC is 48, and the half chord AB or BC is 30 ?
2x30=60 the double chord of the half aris
-48 the chord of the whole arc.
64 the length of the arc required. Prol. 12. The chord and height of a segment being given, to find the radius of the circle.
To the square of the half chord, add the square of the versed sine ; divide the sum by twice the versed sine, and the quotient will be the radius of the circle, when it is less than a semicircle.
Ex. The chord AC of a segment ABC being 48 feet, and the persed sine 18 feet, what is the radius of the circle ?
See the diagram to Example Problem 10.
Prob. 13. Given any two paralle. chords in a circle, and ineir Qistance, to find the distance of the greater chord from the centre.
To the square of the distance between the chords, add the square of half the lesser chord. The difference between this sum, and the square of half the greater chord divided by twice the distance of the chords, will give the distance of the centre from the greatest chord,
Ex. Suppose the greater chord CD is 48 feet, and the lesser AB 30, their distance FG 13 feet, how far is the centre E from the greater chord CD?
G [chord 225 square of the lesser 576 sq. of the greater chord 169 square of the dist. -394
2x 13=26) 18217=EF dist. required.