4. QUADRILATERALS.

347. A Quadrilateral is a plane figure bounded by four straight lines.

348. A Parallelogram is a quadrilateral whose opposite sides are equal and parallel.

349. A Rectangle is a parallelogram whose angles are right angles.

350. A Square is a rectangle whose sides are equal.

351. A Rhombus is a parallelogram whose sides are all equal, but its angles are not right angles.

352. A Rhomboid is a parallelogram whose opposite sides only are equal, but whose angles are not right angles.

SQUARE.

RECTANGLE.

RHOMBUS.

353. A Trapezoid is a quadrilateral only two of whose sides are parallel.

354. A Trapezium is a quadrilateral none of whose sides are parallel.

355. The altitude of a parallelogram, or of a trapezoid, is the perpendicular distance between the parallel sides.

RHOMBOID.

The following principles have been established by Geometry: 1. The area of any parallelogram is equal to the product of its base and altitude.

2. The area of a trapezoid is equal to one-half the sum of its parallel sides, multiplied by its altitude.

1. What is the area of a room 15 ft. long, 9 ft. 6 in. wide? Ans. 1424 sq. ft. 2. A board 16 ft. long is 12 in. wide at one end, and 10 in. wide at the other: how many square feet does it contain? Ans. 143 sq. ft.

3. A field 900 ft. long is 720 ft. wide at one end, and 600 ft. at the other: how many acres does it contain? Ans. 137 A.

NOTE.-A trapezium may be divided by a diagonal into two triangles, and the area of these be found by the rule for finding the area of triangles.

4. The sides of a field are 30, 40, 60 and 70 rods, respectively, and the diagonal joining the two opposite angles 50 rods: what is the area of the field? Ans. 12 A. 149.7 P.

5. CIRCLES.

356. A Circle is a plane figure bounded by a curved line called the circumference, every point of which is equally distant from a point within, called the centre.

357. The diameter of a circle is a straight line passing through the centre and ending on both sides in the circumference.

358. The radius is a line drawn from the centre to the circumference.

The following are the most important rules relating to the circle:

RULES.

1. To find the circumference of a circle, multiply the diameter by 3.1416.

2. To find the diameter of a circle, divide the circumference by 3.1416, or multiply the circumference by .3183.

PROBLEMS.

1. What is the circumference of a circular plot of ground 25 ft. in diameter ? Ans. 78.54 ft.

2. The circumference of a tree is 12.5664 ft.: what is the diameter ? Ans. 4 ft. 3. A circular race-course is 1 mi. in extent: what is the diameter of the enclosure? Ans. 1680.67 ft. 3. The area of a circle the circumference diameter; or, radius squared 3.1416; or, diameter squared .785398. 4. What is the area of a circle whose circumference is 1 mi.? Ans 50 A. 148 P. 194.4-sq. ft.

5. What is the area of a circle mi. in circumference? Ans. 12 A. 117 P. 48.6-sq. ft.

6. What is the area of a circle whose radius is 100 ft.?

Ans. 31416 sq. ft.

4. To find the side of an inscribed square when the diameter of a circle is given-1. Multiply the diameter by .7071; or, 2. Find the square root of half the square of the diameter.

7. How large a square piece of lumber can be cut from a log 3 ft. in diameter ? Ans. 25,4556 in. 8. How large a square piece of lumber can be cut from an 18-in. log? Ans. 12.7278 in.

6. VOLUMES.

359. A Solid or Volume is a body which has three dimensions-length, breadth and thickness.

360. A Prism is a volume whose ends are equal and parallel polygons, and its sides are parallelograms. These sides are called faces, and the ends bases.

361. A prism is triangular, rectangular, square, etc., according to the name of its bases.

362. A prism bounded by six parallelograms, the opposite ones being equal and parallel, is called a Parallelopipedon.

363. A parallelopipedon whose faces are all squares is called a Cube.

364. A Cylinder is a prism whose ends are circles, and whose body is bounded by a uniformly curved surface.

RULES.

1. To find the convex surface of a prism or a cylinder, multiply the perimeter of its base by the altitude.

2. To find the entire surface of a prism or a cylinder, to its convex surface add the area of its bases.

PROBLEMS.

1. What is the convex surface of a log 30 in. in diameter and 10 ft. long? Ans. 78.54 sq. ft.

2. What is the entire surface of a stick of timber 20 ft. long, 2 ft. wide and 1 ft. thick? Ans. 146 sq. ft.

3. What is the convex surface of a triangular prism whose altitude is 12 ft. and its sides 12, 15 and 18 in. respectively? Ans. 45 sq. ft.

4. What is the entire surface of a cylinder 30 ft. long and 40 in. in diameter ? Ans. 331.6133 sq. ft. 5. What will it cost to paint a rectangular prism 6 ft. high whose sides are 30 in. each, one base and the sides only to be painted, at 5 cts. a square foot? Ans. $3.31.

3. To find the contents of a prism or cylinder, multiply the area of the base by the altitude.

6. What are the contents of a stick of timber 60 ft. long, 18 in. wide and 15 in. thick? Ans. 112 cu. ft.

7. What are the contents of a cylindrical column 2 ft. in diameter and 30 ft. high? Ans. 147.2621+ cu. ft.

8. What is the value of a log 50 ft. long whose thickness is 24 in., at 20 cts. a cubic foot?

Ans. $31.416.

9. What are the contents of a cylindrical water-tank whose height is 20 ft. and diameter 30 ft. ?

Ans. 105753.6 gal.

7. PYRAMIDS AND CONES.

365. A Pyramid is a volume whose base is a polygon, and whose sides are triangles terminating in a common point called the vertex.

366. A Cone is a volume having a circular base, and whose surface slopes uniformly to the

vertex.

367. The altitude of a pyramid or a cone is the perpendicular distance from the vertex to the base.

368. The slant height of a pyramid or a cone is the perpendicular distance from the vertex to the perimeter of the base.

NOTE. In a right pyramid the slant height is the altitude of each of the equal triangles which compose the convex surface.

369. The Frustum of a cone or a pyramid is the part which remains after cutting off the top by a plane parallel with the base.

The following are the principles applying to pyramids and cones:

PRINCIPLES.-1. The convex surface of a cone or a pyramid equals the perimeter multiplied by one-half the slant height.

2. The volume of a cone or a pyramid equals the area of the base multiplied by one-third the altitude.

3. The convex surface of the frustum of a cone or a pyramid equals the sum of its perimeters multiplied by one-half the slant height.

4. The volume of the frustum of a cone or a pyramid equals the sum of the areas of the two bases, plus the square root of