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one of the polygons, od radius of inscribed circle to the other polygon (84).

Again, since each side of a regular polygon subtends the same angle at the centre of the inscribed and circumscribing circle, ‹ AOB = ‹ aob, being angles which are the same part of 4 right angles.

L

= 4

Also, since 40=BO, and ao = bo, OAB OBA, and oab: = ▲ oba; but ≤OAB÷‹OBA + ‹ AOB = two right angles = oab + 2 oba + ≤ aob, .. ¿OAB = 4 oab, and ¿OBA = 2 oba, .. OAB and oab are similar triangles. Hence AB: ab :: OA: oa, or :: OD: od; and every pair of sides is in the same ratio; therefore (80)

sum of the sides of one polygon: sum of the sides of the other :: OA: oa, or :: OD: od, that is, the perimeters of the polygons are as the radii of the inscribed or circumscribing circles.

(2) Again, since the polygons are made up of the same number of similar triangles, as AOB, aob; and since AOB: aob :: square of 40: square of ao,

or :: square of OD: square of od,

.. sum of these triangles in one polygon : sum of them in the other :: square of 40: square of ao,

or :: square of OD : square of od;

that is, the areas of the polygons are as the squares of the radii of the inscribed or circumscribing circles.

92. PROP. VII. The areas of similar polygons are to one another as the squares of any homologous sides, or corresponding lines within the polygons.

Let ABCDEF, abcdef be two similar polygons, of which AB, ab are any two corresponding sides; then area ABCDEF: abcdef :: square of AB : square of ab. From A, a, draw the diagonals AC, AD, AE, ac, ad, These will divide the polygons into the same number of triangles, similar and similarly situated, each to each, see fig. (89).

ae.

.. by (76),

triangle ABC triangle abc :: square of AB : square of ab,

ACD:

ADE:

......

......

acd: square of CD: square of cd, ade: square of DE: square of de,

AEF: .. aef :: square of EF: square of ef.

......

But AB: ab :: BC: bc :: CD: cd :: DE: de :: EF: ef (71), of cd: square of AB: square of ab,

of CD: : square

...square
square of DE:

: square of de ::

[blocks in formation]

.. (80) ABC + ACD+ADE+AEF : abc + acd + ade +aef :: square of AB: square of ab, or area ABCDEF: abcdef :: square of AB : square of ab. Again, since AB: ab :: AC: ac :: AD: ad :: AE: ae, .. area ABCDEF: abcdef :: square of AC, or AD, or AE

: square of ac, or ad, or ae, respectively.

93. PROP. VIII. The circumferences of circles are to one another as their radii, or diameters; and their areas are proportional to the squares of those radii, or diameters.

Suppose any two similar regular polygons to have their circumscribing circles drawn about them; these circles will represent any two circles. Bisect each of the arcs subtended by each of the sides of the two polygons, and join the points of bisection with the adjacent angular points of the polygons; then two polygons of double the number of sides will be formed, while the circumscribing circles remain the same; and the perimeters and areas of these latter polygons will obviously approach nearer to the perimeters and areas of the circles than those of the former polygons. Again the arcs subtended by the sides of these polygons may be bisected, and other polygons described with double the number of sides, while the circles remain the same; and so on without limit, until the polygons are made to approach as near as we please to the circles.

Now the perimeters of similar regular polygons are as the radii of their circumscribing circles, and the areas as the squares of those radii, whatever be the number of sides, and therefore when that number, as above, is supposed to be indefinitely increased. But, by thus increasing the number of sides the polygons may be made to differ from the circles by less than any assignable magnitude, both as to perimeter and area. Hence the perimeters, that is, the circumferences of the circles will be as their radii, and the areas as the squares of those radii.

Also, since the diameters will obviously have the same ratio to each other as the radii, the circumferences of circles will be as their diameters, and the areas as the squares of those diameters.

COR. Since circumf. of one circle: circumf. of another diameter of the former: diameter of the latter, .. alternately, circumf. of one : its diameter :: circumf. of the other its diameter; that is, the ratio of the circumference of every circle to its diameter is the same.

:

EXERCISES D.

(1) Define 'hexagon,' and 'diagonal' of a polygon. How many different diagonals has the hexagon?

(2) Define 'angle of a polygon'; and shew that in every polygon the sum of all the angles is a multiple of a right angle.

(3) Shew that the angle of a regular polygon is always greater than a right angle; and that it increases. as the number of sides increases.

(4) Shew that the angle of a regular octagon is equal to one right angle and a half. Hence construct a regular octagon upon a given straight line.

(5) Shew that the side of a regular hexagon is equal to the radius of the circumscribing circle.

(6) What is the number of diagonals which may be drawn in a polygon of ten sides?

(7) Dividing a polygon by means of certain diagonals into the triangles of which it may be supposed to be made up, shew that the number of these triangles will always be less by 2 than the number of sides of the polygon.

(8) Shew that in a regular pentagon each diagonal is parallel to a side; and that, if all the diagonals be drawn another regular pentagon will be formed by their intersections within the former one.

(9) Shew that every regular polygon may be divided into equal isosceles triangles. For what polygon are these triangles equilateral?

(10) Shew that the sum of all the angles of a polygon is not altered by altering the sides either in magnitude or relative position, as long as their number remains the

same.

(11) Having given a regular polygon of any number of sides, shew how a regular polygon of double the number of sides may be constructed.

(12) State the process by which the area of any polygon may be converted into an equivalent rectangle.

(13) Shew that two similar polygons are equal to one another, if a side of the one be equal to the corresponding side of the other.

(14) If two similar polygons be constructed such that a side of the one is ten times the corresponding side of the other, what proportion will the areas of the two polygons have to each other?

(15) If you wished to increase a garden, which is in the form of a polygon, so as to become exactly four times as large as it is, but to retain its present shape, how would you proceed to lay out the boundary?

(16) Can a circle be made which shall have its circumference exactly equal to the circumferences of two other given circles taken together? If so, shew how it may be done.

(17) If the area of one circle be nine times that of another, what is the ratio of their diameters?

(18) Describe a circle whose circumference shall be exactly twice the circumference of a given circle.

(19) Describe a circle whose areu shall be exactly twice the area of a given circle.

(20) Shew that the areas of circles are to one another as the squares inscribed in them.

(21) Shew that all regular polygons of the same name are necessarily similar.

(22) The corresponding sides of two similar polygons are in the ratio of a side of a square to its diagonal; find the ratio of the areas of the polygons.

(23) If in any circle four radii be drawn at right angles to one another, and with each of these four radii as diameters circles be described within the former, shew that the areas of the four circles are together equal to that of the original circle.

(24) From a given polygon cut off a similar polygon whose area shall be one-fourth of the original one.

(25) Shew how the square may be found which is equal to any given polygon.

END OF PART I.

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