3. How many square feet in a board 16 ft. long, 18 inches wide at one end and 25 inches wide at the other end?

4. One side of a quadrilateral field measures 38 rods; the side opposite and parallel to it measures 26 rods, and the distance between the two sides is 10 rods. Find the area.

900. To find the area of a trapezium.

1. Find the area of a trapezium whose diagonal is 42 feet and perpendiculars to this diagonal, as in the diagram, are 16 feet and 18 feet.

18 ft.

42 ft.

OPERATION. (18 ft.+ 16 ft.÷2) × 42 = 714 sq. feet, area.

16 ft.

2. Find the area of a trapezium whose diagonal is 35 ft. 6 in., and the perpendiculars to this diagonal 9 feet and 3 feet.

RULE.-Multiply the diagonal by half the sum of the perpendiculars drawn to it from the vertices of opposite angles.

3. How many acres in a quadrilateral field whose diagonal is 80 rd. and the perpendiculars to this diagonal 20.453 and 50.832 rd. ?

To find the area of any regular polygon, multiply its perimeter, or the sum of its sides, by one-half the perpendicular falling from its center to one of its sides. To find the area of an irregular polygon, divide the figure into triangles and trapeziums, and find the area of each separately. The sum of these areas will be the area of the whole polygon.

THE CIRCLE.

901. 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 center.

902. The Diameter of a circle is a line passing through its center, and terminated at both ends by the circumference.

903. The Radius of a circle is a line extending from its center to any point in the circumference. It is one-half the diameter.

PROBLEMS.

904. When either the diameter or the circumference of a circle is given, to find the other dimension of it.

=

1. Find the circumference of a circle whose diameter is 20 inches. OPERATION.-20 in. × 3.1416 62.832 in. = 5 ft. 2.832 in., circum. 2. Find the diameter of a circle whose circumference is 62.832 ft. OPERATION.-62.832 ft.÷3.1416: = 20 ft., diameter.

3. Find the diameter of a wheel whose circumference is 50 feet.

RULE.—1. Multiply the diameter by 3.1416; the product is the circumference.

2. Divide the circumference by 3.1416; the quotient is the diameter. 4. What is the diameter of a tree whose girt is 18 ft. 6 in.? 5. What is the radius of a circle whose circumference is 31.416 ft.? 6. Find the circumference of the greatest circle that can be drawn with a string 14 inches long, used as a radius.

905. To find the area of a circle, when both its diameter and circumference are given, or when either is given.

1. What is the area of a circle whose diameter is 10 feet and circumference 31.416 feet?

OPERATION.-31.416 ft. × 10÷4=78.54 sq. ft., area.

2. Find the area of a circle whose diameter is 10 feet. OPERATION.-10 ft.2 x .7854

=

78.54 sq. feet, area.

3. Find the area of a circle whose circumference is 31.416 feet. OPERATION.-31.416 ft.÷3.1416=10 ft., diam.; (10 ft.)2 × .7854= 78.54 sq. feet, area.

RULES.-To find the area of a circle :

1. Multiply of its diameter by the circumference.

2. Multiply the square of its diameter by .7854.

4. What is the area of a circular pond whose circumference is 200 chains

5. The distance around a circular park is 11 miles. How many acres does it contain?

906. To find the diameter or the circumference of a circle, when the area is given.

1. What is the diameter of a circle whose area is 1319.472 ?

OPERATION.—1319.472÷.7854 = 1680; √/1680 = 40.987+, diam

eter.

2. What is the circumference of a circle whose area is 19.635?

OPERATION—19.635 ÷ 3.1416 = 6.25; √/6.25=2.5, radius ; 2.5 × 2 x 3.1416 15.708, circumference.

=

RULE.-1. Divide the area by .7854 and extract the square root of the quotient; the result is the diameter.

2. Divide the area by 3.1416 and extract the square root of the quotient; the result is the radius The circumference is obtained by Art. 904, 1. Or,

3. Divide the area by .07958 and find the square root of the quotient. 3. The area of a circular lot is 38.4846 square rods. What is its diameter ?

4. The area of a circle is 286.488 square feet. Required the diameter and the circumference.

907. To find the side of an inscribed square when the diameter of the circle is known.

1. What is the side of a square inscribed in a circle whose diameter is 6 rods?

OPERATION.—62 ÷ 2 = 18;./18=4.24 rods, side of square.

2. The diameter of a circle is 200 feet. Find the side of the inscribed square.

RULE.-1. Extract the square root of half the square of the diam eter. Or,

2. Multiply the diameter by 7071.

3. The circumference of a circle is 104 yards. Find the side of the inscribed square.

4. The area of a circle is 78.54 square feet. Find the side of the inscribed square.

908. To find the area of a circular ring formed by two concentric circles.

Ө

1. Find the area of a circular ring, when the diameters of the circles are 20 and 30 feet.

OPERATION.-(30+ 20 × 30 — 20) × .7854 = 392.7 sq. ft., area.

2. Find the area of a circular ring formed by two concentric circles, whose diameters are

7 ft. 9 in. and 4 ft. 3 in.

RULE.-Multiply the sum of the two diameters by their difference, and the product by .7854; the result is the area.

3. Two diameters are 35.75 and 16.25 ft.; find the area of the ring. 4. The area of a circle is 1 A. 154.16 P. In the center is a pond of water 10 rd. in diameter; find the area of the land and of the water.

909. To find a mean proportional between two numbers.

1. What is a mean proportional between 3 and 12?

OPERATION.-/12 × 3 = 6, the mean proportional.

When three numbers are proportional, the product of the extremes is equal to the square of the mean.

RULE-Extract the square root of the product of the two numbers. Find a mean proportional between

2. 42 and 168.

3. 64 and 12.25.

4. $8 and 1.

5. A tub of butter weighed 36 lb: by the grocer's scales; but weighing it in the other scale of the balance, it weighed only 30 pounds. What was the true weight of the butter?

SIMILAR PLANE FIGURES.

910. Similar Plane Figures are such as have the same form, viz., equal angles, and their like dimensions proportional.

All circles, squares, equiangular triangles, and regular polygons of the same number of sides are similar figures.

The like dimensions of circles are their radii, diameters, and circumferences. PRINCIPLES.-1. The like dimensions of similar plane figures are proportional.

2. The areas of similar plane figures are to each other as the squares of their like dimensions. And conversely,

3. The like dimensions of similar plane figures are to each other as the square roots of their areas.

The same principles apply also to the surfaces of all similar figures, such as triangles, rectangles, etc.; the surfaces of similar solids, as cubes, pyramids, etc.; and to similar curved surfaces, as of cylinders, cones, and spheres. Hence,

4. The surfaces of all similar figures are to each other as the squares of their like dimensions. And conversely,

5. Their dimensions are as the square roots of their surfaces.

PROBLEMS.

1. A triangular field whose base is 12 ch. contains 2 A. 80 P. Find the area of a field of similar form whose base is 48 chains. OPERATION.-122: 482 :: 2 A. 80 P.: x P.-6400 P. (PRIN. 2.)

=

40 A., area.

2. The side of a square field containing 18 acres is 60 rods long. Find the side of a similar field that contains as many acres.

OPERATION.—18 A. : 6 A. : : 602 : x2=1200; √/1200 = 34.64 rd. +, side. (PRIN. 3.)

3. Two circles are to each other as 9 to 16; the diameter of the less being 112 feet, what is the diameter of the greater?

OPERATION.-9: 16:: 1122 x23: 4 :: 112 x 149 ft. 4 in., diameter. (PRIN. 2.)

4. A peach orchard contains 720 square rods, and its length is to its breadth as 5 to 4; what are its dimensions?

OPERATION. The area of a rectangle 5 by 4 equals 20 (898).

20 720 52: x2 = 900;

90030 rd., length.

20 720 42 : x2 =

576;

576 24 rd., width.

5. It is required to lay out 283 A. 107 P. of land in the form of a rectangle, so that the length shall be 3 times the width. Find the dimensions.

6. A pipe 1.5 in. in diameter fills a cistern in 5 hours; find the diameter of a pipe that will fill the same cistern in 55 min. 6 sec.

7. The area of a triangle is 24276 sq. ft., and its sides in proportion to the numbers 13, 14, and 15. Find the length of its sides in feet