8. The length of a rhombus is 17 ft., and its perpendicular height 16 ft.; what is its area? Ans. 272 sq. ft.

9. What is the area of a rhomboid whose altitude is 25 rods, and its length 28.6 rods?

822. To find the Area of a Trapezoid, when its Parallel Sides and Altitude are given.

1. Find the area of a trapezoid whose parallel sides are 28 and 36 feet and its altitude 12 feet.

SOLUTION.-The sum of the parallel sides 28 +36= 64 ft., of 64 = 32 ft., and 32 ft. × 12 (the altitude) = 384 sq. ft., Ans. Hence, the

RULE.-Multiply half the sum of the parallel sides by the altitude.

2. The parallel sides of a trapezoid are 25 yd. and 21 yd., and its altitude 16 yd.; what is its area?

3. Find the area of a trapezoid whose parallel sides are 25 rods and 37 rods, and its altitude 18 rods.

823. To find the Area of a Trapezium, when the Diagonal and Perpendiculars are given.

1. A man bought a city lot in the form of a trapezium, the diagonal of which was 84 ft. and perpendiculars from the opposite angles 12 ft. and 16 ft.; what was its area?

SOLUTION.-The sum of the perpendiculars is

28 ft.; of 28 14 and 84 ft. x 14 1176 sq. ft., Ans. Hence, the

H

=

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

2. A man bought a meadow in the form of a trapezium, the diagonal of which was 250 rods, and the perpendiculars 30 and 35 rods; how many acres did it contain?

3. Find the area of an irregular piece of land, the diagonal of which is 320 yards, and the perpendiculars 35.5 yards and 421 yards.

CIRCLES.

824. A Circle is a plane figure bounded by a curve line, every part of which is equally distant from a point within called the center.

825. The Circumference of a circle is the curve line by which it is bounded.

826. The Diameter is a straight line drawn through the center, terminating at each end in the circumference.

827. The Radius is a straight line drawn from the center to the circumference, and is equal to half the diameter.

NOTE. From the definition of a circle, it follows that all the radii are equal; also, that all the diameters are equal.

828. From the relation of the circumference and diameter to each other, we derive from Geometry the following

PRINCIPLES.

1°. The Circumference the Diameter x 3.1416 nearly. 2°. The Diameter of a Circle = the Circumference÷3.1416 nearly.

829. To find the Circumference of a Circle, when the Diameter is given.

1. What is the circumference of a circle whose diameter is 15 feet?

SOLUTION.-15 ft. x 3.1416 = 47.125 ft., Ans. Hence, the

RULE. Multiply the given diameter by 3.1416. (Art. 828, 1°.)

2. What is the circumference of a circle whose diameter is 45 yards?

3. What is the circumference of a circle whose diameter is 100 rods?

830. To find the Diameter of a Circle, when the Circumference

is given.

1. What is the diameter of a circle whose circumference is 65 feet?

SOLUTION.-65.5÷3.1416

20.849+ ft., Ans. Hence, the

RULE.-Divide the circumference by 3.1416. (Prin. 2°.)

2. What is the diameter of a circle whose circumference is 94.2477 rods ?

3. What is the diameter of a circle whose circumference is 628.318 yards?

NOTE.-The diameter of a circle may also be found by dividing the area by .7854, and extracting the square root of the quotient.

4. Required the diameter of a circle containing 50.2656 square rods.

5. Required the diameter of a circle containing 201.0624 square feet.

831. To find the Area of a Circle, when the Diameter and Circumference are given.

1. What is the area of a circle whose diameter is 20 ft. and circumference 31.416 ft.?

SOLUTION.-31.416 x 20 157.08 sq. ft., Ans.

Or, 31.416 × (20÷4) = 157.08 sq. ft., Ans. Hence, the

RULE.—Multiply half the circumference by half the diameter; or,

Multiply the circumference by a fourth of the diameter.

NOTES.-1. If only one of these dimensions are given, the other must be found before the rule can be applied. (Ex. 3, 4.)

2. The area of a circle may also be found by multiplying the square of its diameter by the decimal .7854.

2. Find the area of a circle whose diameter is 20 ft.?

3. What is the area of a circle whose diameter is 100 ft.? 4. What is the area of a circle whose diameter is 120 rods?

5. What is the area of a circle whose circumference is

160 yards?

6. What is the diameter of a wheel whose circumference is 50 ft.?

7. Find the circumference of a tree whose diameter is 3 ft. 4 in.

8. What is the area of a circle whose radius is 15 ft.?

9. How many acres in a circular park whose circumference is 2 miles?

10. What is the radius of a circle which contains 14 acre ?

SOLIDS.

832. A Solid is that which has length, breadth, and thickness.

833. A Prism is a solid whose bases are similar, equal, and parallel, and whose sides are parallelograms.

NOTE.-Prisms are named from the form of their bases, as triangular, quadrangular, pentagonal, hexagonal, etc.

834. A Right Prism is one whose sides are perpendicular to its bases.

835. A Triangular Prism is one whose bases are triangles.

836. A Rectangular Prism is

one whose bases are rectangles, and its sides perpendicular to its bases.

837. The Lateral Surface of a prism is the sum of all its faces.

838. The Altitude of a prism is the perpendicular distance between its bases.

839. All rectangular solids are prisms.

NOTES.-1. When their sides are all equal to each other they are called cubes.

2. When their bases are parallelograms they are called parallelopipeds, or parallelopipedons.

840. A Cylinder is a circular body of uniform diameter, whose ends are equal parallel circles.

841. To find the Lateral Surface of a Prism or Cylinder.

1. What is the lateral surface of a prism whose altitude is 12 ft., and its base a pentagon each side of which is 6 feet? SOLUTION.-6 ft. x 530 ft. the perimeter.

30 ft. x 12 = 360 sq. ft. surface, Ans.

2. What is the convex surface of a cylinder 32 inches in circumference and 40 inches long?

SOLUTION.-32 × 40

=

1280 sq. in., Ans. Hence, the

RULE.-Multiply the perimeter of the base by the

altitude.

NOTE.-To find the entire surface, the area of the bases must be added to the lateral surface.

3. What is the surface of a triangular prism wh is 9 feet, and the sides of its base are 3, 4, and tively?

4. Required the lateral surface of a triangular perimeter is 4 inches, and its length 12 inches.

5. Required the lateral surface of a quadrans whose sides are each 2 feet, and its length 19 feet.

6. Required the convex surface of a log whose circumference is 18 ft., and length 32 ft.?

7. What is the convex surface of a cylinder 16 feet in circumference and 40 feet long?

8. What is the convex surface of a cylinder whose diameter is 20 feet and its height 65 feet?