Plane trigonometry, by S.L. Loney.

XXXIX.]

REGULAR POLYGONS.

259

16. There are two regular polygons the number of sides in one being
double the number in the other, and an angle of one polygon is to an angle
of the other as 9 to 8; find the number of sides of each polygon.
17. Show that there are eleven pairs of regular polygons such that
the numlber of degrees in the angle of one is to the number in the angle of
the other as 10: 9. Find the number of sides in each.
18. The side of a base of a square pyramid is a feet and its vertex is
at a height of Ih feet above the centre of the base; if 0 and 0 be respectively the inclinations of any face to the base, and of any two faces to one
another, prove that
2h 5 2
tan =- and tan - = /+
~a 2 l 2h2
19. A pyramid stands on a regular hexagon as base. The perpendicular from the vertex of the pyramid on the base passes through the
centre of the hexagon and its length is equal to that of a side of the base.
Find the tangent of the angle between the base and any face of the
pyramid and also of half the angle between any two side faces.
20. A regular pyramid has for its base a polygon of t sides, each of
length a, and the length of each slant side is I; prove that the cosine of
the angle between two adjacent lateral faces is
2wr
412 cos - + a2
n
412 - a2

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