being 482 feet, the shorter 324 feet, and the altitude 216 feet? Ans. 87048 square feet.

2. What is the area of a plank, whose length is 22 feet, the width of the wider end being 28 inches, and of the narrower 20 inches? Ans. 44 square feet.

ART. 334. A TRAPEZIUM is a quadrilateral, which has neither two of its opposite sides parallel.

A diagonal is a line joining any two opposite angles of a quadrilateral; as EF.

ART. 335. To find the area of a trapezium.

RULE.

E

Divide the trapezium into two triangles by a diagonal, and then find the areas of these triangles; their sum will be the area of the trapezium.

1. What is the area of a trapezium, whose diagonal is 65 feet, and the length of the perpendiculars let fall upon it are 14 and 18 feet? Ans. 1040 square feet.

2. What is the area of a trapezium, whose diagonal is 125 rods, and the length of the perpendiculars let fall upon it are 70 and 85 rods? Ans. 9687.5 square rods.

THE POLYGON.

ART. 336. A POLYGON is any rectilineal figure having more than four sides and four angles. It takes the particular names of pentagon, which is a polygon of five sides; hexagon, one of six sides; heptagon, one of seven sides; octagon, one of eight sides; nonagon, one of nine sides; decagon, one of ten sides; undecagon, one of eleven sides; and dodecagon, one of twelve sides.

ART. 337.

A REGULAR POLYGON is a plane rectilineal figure, which has all its sides and all its angles equal.

The perimeter of a polygon is the sum of all its sides.

ART. 338. To find the area of a regular polygon.

QUESTIONS. -Art. 334. What is a trapezium? What is a diagonal ? — Art. 335. What is the rule for finding the area of a trapezium? - Art. 336. What is a polygon? What particular names does it take?-Art. 337. What is a regular polygon?

-

RULE. Multiply the perimeter by half the perpendicular let fall from the centre on one of its sides, and the product will be the area. 1. What is the area of a regular pentagon, whose sides are each 35 feet, and the perpendicular 24.08 feet?

Ans. 2107 square feet. 2. What is the area of a regular hexagon, whose sides are each 20 feet, and the perpendicular 17.32 feet?

Ans. 1039.20 square feet.

THE CIRCLE.

ART. 339. A CIRCLE is a plane figure bounded by a curved line, every part of which is equally distant from a point, called its centre.

The circumference or periphery of a circle is the line which bounds it.

The diameter of a circle is a line drawn through the centre, and terminated by the circumference; as G H.

ART. 340. To find the circumference of a circle, the diameter being given.

RULE. Multiply the diameter by 3.141592, and the product is the circumference.

1. What is the circumference of a circle, whose diameter is 50 feet? Ans. 157.0796+ feet. 2. A gentleman has a circular garden whose diameter is 100 rods; what is the length of the fence necessary to enclose it? Ans. 314.15+ rods.

ART. 341. To find the diameter of a circle, the circumference being given.

RULE. Multiply the circumference by .318309, and the product will be the diameter.

1. What is the diameter of a circle, whose circumference is 80 miles? Ans. 25.46+ miles. 2. If the circumference of a wheel is 62.84 feet, what is the diameter ? Ans. 20+ feet.

QUESTIONS.- What is the perimeter of a polygon?— Art. 338. What is the rule for finding the area of a regular polygon? Art. 339. What is a circle? What is the circumference of a circle? The diameter of a circle? - Art. 340. What is the rule for finding the circumference of a circle, the diameter being given? Art. 341. What is the rule for finding the diameter of a circle, the circumference being given?

ART. 342. To find the area of a circle, the diameter, the circumference, or both, being given.

RULE 1.- Multiply the square of the diameter by .785398, and the product is the area. Or,

RULE 2.- Multiply the square of the circumference by .079577, and the product is the area.

Or,

RULE 3.- · Multiply half the diameter by half the circumference, and the product is the area.

1. If the diameter of a circle be 200 feet, what is the area? Ans. 31415.92 square feet. 2. There is a certain farm, in the form of a circle, whose circumference is 400 rods; how many acres does it contain? Ans. 79A. 2R. 12+p.

ART. 343. To find the side of a square equal in area to a given circle.

The square in the figure is supposed to have the same area as the circle.

RULE 1.

- Multiply the diameter by .886227, and the product is the side of an equal square. Or,

RULE 2. Multiply the circumference by .282094, and the product is the side of an equal square.

1. We have a round field 40 rods in diameter; what is the side of a square field that will contain the same quantity ?

Ans. 35.44 rods. 2. I have a circular field 100 rods in circumference; what must be the side of a square field that shall contain the same area? Ans. 28.2+ rods. ART. 344. To find the side of a square inscribed in a given circle.

A square is said to be inscribed in a circle when each of its angles touches the circumference or periphery of the circle.

QUESTIONS. Art. 342. What is the rule for finding the area of a circle, when the diameter is given? When the circumference is given? When the diameter and circumference are both given? Art. 343. What is the first rule for finding the side of a square equal in area to a given circle? What the second? Art. 344. When is a square said to be inscribed in a circle? What is the first rule for finding the side of a square inscribed in a circle? The second?

RULE 1.- Multiply the diameter by .707106, and the product is the side of the square inscribed. Or,

RULE 2.

-

Multiply the circumference by .225079, and the product

is the side of the square inscribed.

1. What is the thickness of a square stick of timber that may be hewn from a log 30 inches in diameter ?

2. How large a square field may

circumference is 100 rods?

Ans. 21.21+ inches. be inscribed in a circle whose Ans. 22.5+ rods square.

THE ELLIPSE.

ART. 345. An ELLIPSE is an oval figure having two diameters, or axes, the longer of which is called the transverse and the shorter the conjugate diameter.

ART. 346. To find the area of an ellipse.

RULE. Multiply the two diameters together, and their product by .785398; the last product is the area.

1. What is the area of an ellipse whose transverse diameter is 14 inches, and its conjugate diameter 10 inches?

Ans. 109.95 square inches. 2. What is the area of an elliptical table, 8 feet long and 5 feet wide? Ans. 31 square feet, 59+ square inches.

§ XLIV. MENSURATION OF SOLIDS.

ART. 347. A SOLID is a magnitude which has length, breadth, and thickness.

Mensuration of solids includes two operations: first, to find their superficial contents, and, second, their solidities.

THE PRISM.

ART. 348. A PRISM is a solid whose ends are any plane figures which are equal and similar, and whose sides are parallelograms. It takes particular names, according to the figure

QUESTIONS. Art. 345. What is an ellipse? What is the longer diameter called? The shorter?- Art. 346. What is the rule for finding the area of an ellipse?- Art. 347. What a solid? What two operations does mensuration of solids include? - - Art. 248. What is a prism? What particular names does it take?

of its base or ends, namely, triangular prism, square prism, pentagonal prism, &c.

The base of a prism is either end; and of solids in general, the part upon which they are supposed to stand.

All prisms whose bases are parallelograms are comprehended under the general name parallelopipedons or parallelopipeds.

A triangular prism is a solid whose base is a triangle.

A square prism is a solid whose base is a square, and when all the sides are squares it is called a cube.

A pentagonal prism is a solid whose base is a pentagon.

ART. 349. To find the surface of a prism.

RULE. Multiply the perimeter of its base by its height, and to this product add the area of the two ends; the sum is the area of the prism. 1. What are the superficial contents of a triangular prism, the width of whose side is 3 feet, and its length 15 feet?

Ans. 142.79+ square feet.

2. What is the surface of a square prism, whose side is 9 feet Ans. 1062 square feet.

wide, and its length 25 feet?

ART. 350. To find the solidity of a prism.

RULE.

Multiply the area of the base by the height, and the product is the solidity.

QUESTIONS.-What is the base of a prism and of solids in general? What is a parallelopiped or parallelopipedon? What is a triangular prism? A square prism? A pentagonal prism?1 Art. 349. What is the rule for finding the surface of a prism?-Art. 350. What is the rule for finding the solidity of a prism?