9. Three sides of a quadrilateral are 20 rds., 30 rds., 40 rds., the angle included by the first and second is 60°, and between the second and third, 80°; required the area. Ans. 593.58 sq. rds.

10. The sides of a quadrilateral inscribed in a circle are 40 rds., 50 rds., 60 rds., 70 rds.; required the area. Ans. 2898.28 sq. rds.

11. The area of a parallelogram is 47.055 sq. ft., the sides are 6 ft. and 8 ft.; required the diagonal.

Ans. 9 ft., or 10.906 ft. 12. If the adjacent sides of a parallelogram are- b and c, and their included angle A, find A and k when k is a maximum. Ans. A 90°, k bc.

=

13. The sides and angles being expressed as in the last example, find A and k when k is a minimum. Ans. A=0° or 180°, k 0.

14. If only two adjacent sides, b and c, of a parallelogram be given, prove that k is indeterminate between the limits 0 and bc.

The diagonals divide the polygon into triangles whose sides are given.

15. Prove that the diagonals of a parallelogram divide it into four equal triangles.

The areas of these triangles, k', k", k"", are found by article 160, (5).

=

164. Problem.

To find the area of an irregular polygon.

1. When the sides and diagonals from the same vertex are given.

(1) k k' + k" + k""+...

k

2. When the diagonals from the same vertex, and the perpendiculars to these diagonals from the opposite vertices are given.

(2) k= dp + 1⁄2 d′p′+žď′p′′+ ...

3. When the perpendiculars to a diagonal from the vertices of the opposite angles and the segments of the diagonal made by these perpendiculars are given.

The polygon is divided into right triangles and trapezoids, whose areas k', k", k'", .... are found by article 162, (2), (4). (3) k k' + k" + k”" +

...

If a' a and a" a, (4) becomes,

4. When one side of a figure is a straight line, and the opposite side is an irregular curve or broken line.

=

=

Let the straight line be divided

into the parts a, a', a", ...., and let the perpendiculars be p, q, r,... dividing the figure into parts which may be con

sidered trapezoids.

.. (4) ka (p + q) + } a′(q + r) + 3 a′′ (r + 8).

165. Examples.

1. Find the area of the annexed polygon if p = 10 rds., q= 6 rds., 1= 6 rds., 8 = 15 rds.,

7 rds., t

d

14 rds., d'

16 rds.

=

(5) ka(p + 2 g + 2r+ 8).
q

p

d

sq. rds.

2. Find the area of the annexed polygon if p = 3 rds., d 9 rds., p' 4 rds., d' 12 rds.,

and

p"

5 rds.

Ans. 67.5 sq. rds.

3. Find the area of the annexed polygon if p3 ft., p' 5 ft., p"5 ft., 6 ft., c = 6 ft.,

=

b

4 ft., a
d 9 ft., e=

8 ft.

Ans. 80.5 sq. ft.

k

4. Find the area

q

figure. p 2 rds.,
4 rds., 83 rds., aa'

k

p

of

as +

· a (s + 8 +8 +8 + ...)

.. (1) k παρ.

S. N. 14.

=

the annexed

3 rds., r =

a" 5 rds.

=

: =

DOB

166. Problem.

To find the area of a regular polygon.

1. When the perimeter and apothegm are given.

Let p be the perimeter, a the apo

them, and s one side of the polygon.

Lets be one side, n the number of sides, a the apothem, and p the perimeter.

as+as+as+...

p

DB cot DOB, or a =

2. When the value of each side and the number

of sides are given.

scot

p"

Ans. 47.5 sq. rds.

a'

A

180°

n

a'

D B

If s

(2) k

1, then (3) k

=

=

=

ncot

From (3) calculate the areas of the regular polygons each of whose sides is 1, as given in the table. subjoined.

ns2 cot

=

180°

n

180°

n

167. Table.

= 0.4330127.

= 4.8284271.

Triangle
Square
Pentagon 1.7204774.

1.0000000.

= 6.1818242.

7.6942088.

2.5980762.

Hexagon
Hendecagon 9.3656399.
Heptagon 3.6339124. Dodecagon

11.1961524.

Octagon

Enneagon

Decagon

.. k

168. Application of the Table.

Denoting the area of a regular polygon whose side is s by k, and the area of a similar polygon whose side is 1, as given in the table by k', and applying the principle that the areas of similar polygons are to each other as the squares of the homologous sides, we have the proportion,

kk' s2: 12.

169. Examples.

1. What is the area of a regular whose sides is 6?

=

=

k's2.

hexagon each of Ans. 93.5307432.

2. What is the area of a regular pentagon each of whose sides is 10? Ans. 172.04774.

3. What is the area of a regular decagon each of whose sides is 20? Ans. 3077.68352.

4. What is the area of a regular dodecagon each of whose sides is 100? Ans. 111961.524.

5. What is the area of a regular enneagon each of whose sides is 30? Ans. 5563.64178.

170. Formulas for the Circle.

Let r be the radius, d the diameter, c the circumference, and k the area of a circle, then, by Geometry, we have

1.

2.

3.

πd, k

From which verify the following table of formulas:

r =

r = 1 d.

r =

4. d

5. d

6. d

d

=

=

=

= 2

k

π

2 r, с ==

k

π

7.

= rc.

C 2 r.

8. c = ad.

171. Examples.

1. Given the radius of a circle 10 rds.; required d, c, and k.

2. Given the diameter of a circle 20 rds.; required r, c, and k.