3. Given the circumference of a circle = 150 rds.; required r, d, and k.

1000 sq. rds.; re

4. Given the area of a circle

quired r, d, and c.

5. Find the diameter of a circle whose area is equal to that of a regular decagon, each side of which is 10 ft. Ans. 31.3.

6. The radius of a circle is 10 ft., the diagonals of an equal parallelogram are 24 ft. and 30 ft.; required their included angle. Ans. 60° 46′ 17′′.

7. The radii of two concentric circles are r and r'; find the area of the ring included by their circumferences. Ans. (r) (r — r').

172. Problem.

To find the area of a sector of a circle. Let a be the arc of a sector, d the degrees in the arc, r the radius, and k the

area.

(2) a =

ra.

dar

180

(3) k

173. Examples.

1. Find the area of a sector whose arc is 40° and radius is 10 ft.

Ans. 34.907 sq. ft.

2. Find the area of a sector whose arc is 60° 24′ 30′′ and radius is 100 rds. Ans. 5271.64 sq. rds.

3. The area of a sector is 345 sq. ft., the radius is 20 ft.; required the arc. Ans. 98° 50′ 06′′.

4. The area of a sector is 1000 sq. rds., the arc is 30° 45'; required the radius. Ans. 61.04 rds.

174. Problem.

To find the area of a segment of a circle. Let d be the degrees in the arc of the segment, r the radius, and k the

area.

ત

If d is greater than 180, sin d is negative, and the second term in the value of k becomes positive, as it should, since, in this case, the segment is equal to the corresponding sector plus the triangle.

175. Examples.

1. Find the area of the segment of a circle whose arc is 36° and radius 10 ft. Ans. 2.027 sq. ft.

2. Find the area of a segment whose chord is 36 ft. and radius 30 ft. Ans. 147.30 sq. ft.

3. Find the area of a segment whose altitude is 36 rds, and radius 50 rds. Ans. 2545.85 sq. rds.

4. The arc of a segment is 147°28'46", the altitude is 36 rds.; required the area.

Ans. 2545.85 sq. rds.

176. Problem.

To find the area of an ellipse.

Let a be the semi-major axis, and b the semi-minor axis.

Then, Ray's Analytic Geometry, article 446,

πab.

2 k's2

177. Examples.

1. The semi-axes of an ellipse are 10 in. and 7 in.; required the area. Ans. 219.912 sq. in. 2. The area of an ellipse is 125 sq. rds.; find the axes if they are to each other as 3 is to 2.

Ans. 15.45; 10.30.

k

178. Problem.

To find the area of the entire surface of a right prism.

Let p be the perimeter of the base,

a the altitude, s one side of the base, k' the area of a polygon similar to the base, each side of which is unity, article 167, and k the area of the entire surface.

=

k

the areas of the bases. Article 168.

ap + 2 k's2.

u

179. Examples.

1. What is the entire surface of a right prism whose altitude is 20 ft., and base a regular octagon each side of which is 10 ft.? Ans. 2565.68542 sq. ft.

2. What is the entire surface of a right hexagonal prism whose altitude is 12 ft., and each side of the base is 6 ft.? Ans. 619.0614864 sq. ft.

3. What is the entire surface of a right prism whose altitude is 15 in., and base a regular triangle each side of which is 3 in.? Ans. 142.7942286 sq. in.

ξαρ
k's 2

180. Problem.

To find the area of the surface of a regular pyramid.

Let p be the perimeter of the base, a the slant height, s one side of the base, k' and k as in the last problem.

the convex surface.

the area of the base.

181. Examples.

1. What is the entire surface of a regular pyramid whose slant height is 12 ft., and base a regular triangle each side of which is 5 ft.?

Ans. 100.82532 sq. ft.

2. What is the entire surface of a right pyramid whose slant height is 100 ft., and base a regular decagon each side of which is 20 ft.?

Ans. 13077.68352 sq. ft.

182. Problem.

To find the entire surface of a frustum of a right pyramid.

Let p be the perimeter of the lower base, p' the perimeter of the upper base, a the slant height, s one side of the lower base, s' one side of the upper base, k' and k as in Art. 178.

the convex surface.

a(p+p')
k's 2
k's'2= the area of upper base.

the area of lower base.

... k = a (p + p') + k' (82 + 8′2).
= ≥

=

183. Examples.

1. What is the entire surface of a frustum of a pyramid whose slant height is 12 ft., and the bases regular decagons whose sides are 8 ft. and 5 ft., respectively? Ans. 1464.78458 sq. ft.

2 πγα =

2 πr2

2. What is the entire surface of a frustum of a pyramid whose slant height is 15 ft., and the bases regular hexagons whose sides are 10 ft. and 6 ft., respectively? Ans. 1073.338 sq. ft.

184. Problem.

To find the area of the entire surface of a cylinder. Let be the radius of the cylinder, a its altitude, and k the area of the entire surface.

the convex surface.

the area of the bases.

... k2 πr (a + r).