Plane trigonometry, by S.L. Loney.

258

TRIGONOMETRY.

[Exs.

5. A square, whose side is 2 feet, has its corners cut away so as to
form a regular octagon; find its area.
6. Compare the areas and perimeters of octagons which are respectively inscribed in and circumscribed to a given circle, and shew that the
areas of the inscribed hexagon and octagon are nearly as ^/27 to ^/32.
7. Prove that the radius of the circle described about a regular
pentagon is nearly oIths of the side of the pentagon.
8. If an equilateral triangle and a regular hexagon have the same
perimeter, prove that their areas are as 2: 3.
9. If a regular pentagon and a regular decagon have the same
perimeter, prove that their areas are as 2: ^/5.
10. Prove that the sum of the radii of the circles, which are respectively inscribed in and circumscribed about a regular polygon of n sides, is
a 7r
a cot '
2 2n'
where a is a side of the polygon.
11. Of two regular polygons of n sides, one circumscribes and the
other is inscribed in a given circle. Prove that the three perimeters are in
the ratio
7r 7r 7r
sec -: - cosec -: 1,
n n i
and that the areas of the polygons are in the ratio cos2 -: 1.
n
12. Given that the area of a polygon of n sides circumscribed about
a circle is to the area of the circumscribed polygon of 2n sides as 3: 2,
find n.
13. Prove that the area of a regular polygon of 2n sides inscribed in a
circle is a mean proportional between the areas of the regular inscribed
and circumscribed polygons of n sides.
14. The area of a regular polygon of n sides inscribed in a circle is to
that of the same number of sides circumscribing the same circle as 3 is to
4. Find the value of n.
15. The interior angles of a polygon are in A..; the least angle
is 120~ and the common difference is 5~; find the number of sides.