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To find the area of the trapezium :—
Divide it into two triangles by a diagonal line, (as ab in the figure,) to which draw perpendiculars from the opposite angles; measure the length of the diagonal and the two perpendiculars, and multiply the diagonal by the sum of the perpendiculars, and half the product is the area. Or, Find the areas of the two triangles as directed in ¶ 19, and the sum of their areas will be the area of the trapezium.
1. What is the area of a trapezium, whose diagonal is 42, and the perpendiculars falling upon it from the opposite angles 16 and 18? Ans. 714.
2. How many square yards of paving are there in a trapezium whose diagonal is 65, and the perpendiculars falling upon it 28 and 32.5 feet? Ans. 222.08 yards.
When a trapezium can be circumscribed by a circle, and its four angles made to touch the circumference, its area may be found by the following rule.
From the sum of the four sides subtract each side, severally; then multiply together the four remainders and extract the square root of their product.
3. The sides of a trapezium inscribed in a circle, are 10, 12, 15, and 21 rods; its area is required. Ans. 190.2 rods.
4. The four sides of a trapezium in a circle, are 600, 650, 700, and 750 links; its area is required.
Ans. 4 acres, 2 roods, 4 rods.
Any figure bounded by more than four right lines, is called a polygon; and if the sides are of equal length, it is called a regular polygon; but if the sides are of unequal length, it is called an irregular polygon. To find the area of an irregular polygon :
Divide it into triangles; measure each of the triangles sepa
rately and find its area, and the sum of all the triangles will be the area of the polygon.
Find the area of an hexagonal figure from these measurements:
AC 525 links, BG=160 links, DF=490 links, FH-210 links, EI=100 links, CK=300 links.
Ans. 1 acre, 3 roods, 32.2 rods.
The figure, it is evident, is made up of the two trapeziums ABCF, and CDEF.
Required the area of the irregular hexagon ABCDEF from these data :
The side AB=690 links, BC=870 links, CD=770 links, DE=620 links, EF=770 links, AF=630 links, the diagonal FD=1210 links, the diagonal AD=1634 links, and the diagonal BD=1486 links. Ans. 12 acres, 3 roods, 37.19 poles.
In this example the polygon is divided into triangles of which the three sides are known; and hence their areas may be found by the rules for the triangle. See ¶ 19, under problem 10.
The sum of all the angles of any polygon is equal to twice as many right angles, wanting four, as the figure has sides; and when the polygon is regular, if the sum of the angles be divided by the number of angles, the quotient will be the value of one of the angles. Thus
10 × 90°—(4 × 90°)=90.0°— 360°-540°, the sum of the
angles of a pentagon; and =108°, the interior angle of a regular polygon of 5 sides.
21. REGULAR POLYGONS.
A regular polygon has its sides and angles equal; and its area is equal to that of a rectangle, one side of which is equal to the perimeter, or sum of all the sides of the polygon, and the other to one-half of the radius of the greatest inscribed circle, or to half the distance from the centre of the polygon to the middle of one of its sides. Therefore, to find the area of any regular polygon:
Multiply the perimeter of the polygon by half the radius of the greatest inscribed circle.
A regular polygon of six sides is called a hexagon; one having five sides is called a pentagon; one having seven sides is called a heptagon; one having eight sides, an octagon; one having nine sides, a nonagon; one having ten sides, a decagon;
one having eleven sides, an undecagon; and one having twelve sides is called a dodecagon.
To draw a pentagon:-Draw the side ab of any required length; then take the side ab between the dividers, and having set one foot on the end of the line, at a, strike a semicircle, and divide the arc into five equal parts; then from the end of the line at a, draw a line through the first space from b, and extend it indefinitely; then with the side ab between the dividers, place one foot on the end of the line at b, and with the other mark the line ac; then place one foot of the dividers on the point found in the line ac at c, and strike an arc of a circle over the base of the polygon; then place one foot of the dividers on the third space from 6 on the arc of the semicircle, and strike an arc over the base of the polygon cutting the arc struck from the point c, and where the two arcs meet will be the vertex of the pentagon; then through the points found as described above, draw the sides of the pentagon.
To find the diagonal of any pentagon :-Multiply the given side by 1.618036. To find the perpendicular altitude of a pentagon:-Multiply the given side by 1.588842. To find the
apothegm of a pentagon, that is, the distance from its centre to the centre of one of its sides:-Multiply the given side by .688191. To find the radius of the least circumscribing circle: -Multiply the given side of the pentagon by .850651.
What is the area of a pentagon whose side is 1?
Any regular polygon may be divided into as many equal triangles as the polygon has sides, by drawing lines from its centre to its several angles; hence it is manifest that we may find its area by multiplying its apothegm by half the perimeter of the polygon; or, by multiplying its perimeter by half its apothegm; that is, by half the perpendicular altitude of one of the several triangles which together make up the polygon.
The area of all regular polygons are as the squares of their sides. Therefore, if we multiply the square of the side of any regular polygon by the area of a similar polygon whose side is 1, it will give the area.
The following table exhibits the areas and apothegms of all regular polygons of not more than 12 sides, when the side is unity:
1. What is the diagonal of a pentagon, whose side is 7?