3. Required the area of a triangular field whose base is 965 rods and altitude 576 rods. Ans. 1737 A.

4. What is the area of a field whose sides are respectively 20, 30, and 40 chains? Ans. 29 A. 8 P.-.

THE QUADRILATERAL.

744. A Quadrilateral is a polygon having four sides and therefore four angles There are three classes, the par.

allelogram, trapezoid, and trapezium.

745. A Parallelogram is a quadrilateral whose opposite sides are parallel. The altitude is the perpendicular distance between its opposite sides.

746. A parallelogram which is right-angled

is called a Rectangle. When the four sides are equal it is called a Square.

747. An oblique-angled parallelogram

is called a Rhomboid.

An equilateral

rhomboid is called a Rhombus.

Rule. To find the area of a parallelogram, multiply the base by the altitude.

1. What is the area of a parallelogram 20 feet long and 18 feet wide? Ans. 40 sq. yd.

2. A has a rectangular lot 192 chains long and 65 chains what is its area? Ans. 1248 acres.

wide; 3. What is the difference in the area of two lots, one being 245 rd. long, 42 rd. wide, and the other 85 chains long and 18 chains wide? Ans. 88 A. 110 P.

748. A Trapezoid is a quadrilateral which has two of its sides parallel. Its altitude is the perpendicular distance between its parallel sides.

Rule. To find the area of a trapezoid, multiply onehalf the sum of the parallel sides by the altitude.

1. Required the area of a trapezoid, one side being 120 in., the other 96 in., and the altitude 48 in. Ans. 36 sq. feet. 2. What is the area of a trapezoid, the sides being 365 and 124 in.. and the altitude 86 in. ? Ans. 146 sq. ft. 3 sq. in.

3. What is the area of a plank 12 feet long, 18 inches wide at one end, and 12 inches at the other end?

Ans. 15 sq. ft.

4. A farmer has a field in the form of a trapezoid, the two parallel sides being 95 and 75 rods respectively, and the perpendicular distance between them being 65 rods; how much land in the field? Ans. 34 A. 85 P.

749. A Trapezium is a quadrilateral which has none of its sides parallel. A diagonal, as AB, divides the trapezium into two triangles.

P

Rule. To find the area of a trapezium, divide the trape zium into two triangles by a diagonal, find the area of each triangle and take the sum.

1. What is the area of a trapezium whose diagonal is 145 in., and the altitudes of the triangles, the diagonal being the base, are 30 and 40 inches respectively?

Ans. 35 sq. ft. 35 sq. in. 2. Required the area of a trapezium, the length of whose sides are respectively 20, 30, 25, and 35 chains, and the length of the diagonal 40 chains.

THE CIRCLE.

Ans. 72 A. 56 P.—.

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

751. The curved line is called the cir.

cumference, and a line passing through the centre and ending in the circumference is the diameter. Half the diameter is called the radius.

752. Rule. To find the circumference of a circle, mul tiply the diameter by 3.1416.

1. What is the circumference of a circle whose diameter is 25 inches? Ans. 78.54 in.

2. What is the distance around a circular fish-pond, the diameter of which is 16 rods? Ans. 50.2656 rà. 3. A man has a garden in the form of a circle, the diameter of which is 45 rods; what is the distance around it? Ans. 141.372 rd

753. Rule.-To find the diameter of a circle, multiply

the circumference by .3183.

1. What is the diameter of a circle whose circumference is 40 feet? Ans. 12.732 feet.

2. What is the diameter of a water-wheel whose circumference is 78.54 feet? Ans. 25 feet.

754. Rule I.- The area of a circle equals the circumference multiplied by one-fourth of the diameter, or the square of the circumference multiplied by .07958.

Rule II.-The area of a circle equals the square of the radius multiplied by 3.1416, or the square of the diameter multiplied by .785398.

NOTE.-The area will vary slightly in the decimal figures as we use the ¿fferent rules.

1. What is the area of a circle whose diameter is 25 and circumference 78.54? Ans. 490.875. 2. What is the area of a circle whose diameter is 36 inches? Ans. 1017.8784 sq. in. 3. What is the area of a circular garden whose circumfer ence is 180 rods? Ans. 2578.23 sq. rd.

755. A square is inscribed in a circle when

each of its angles is in the circumference.

Rule. To find the side of an inscribed square, multiply the diameter by .707106, or multiply the circumference by .225079.

1. What is the side of a square that can be cut out of a circular board whose diameter is 14 inches?

Ans. 9.899 in.

2. How large a square can be cut out of a circular board

whose circumference is 200 inches?

Ans. 45.0158 in.

THE ELLIPSE.

756. An Ellipse is a plane figure bounded by a curved line, the sum of the distances from every point of which to A two fixed points is equal to the line drawn through those points and terminated by the curve. The two fixed

D

points are called foci: the line through the foci is the trans verse axis, and a line perpendicular to this passing through the centre and terminated by the curve, is the conjugate axis.

Rule. To find the area of an ellipse, we multiply half of the two axes together, and that product by 3.1416.

1. What is the area of an ellipse whose transverse axis is 20 inches and conjugate axis is 16 inches?

Ans. 251.328 sq. in.

2. Required the area of an elliptical mirror whose length is 6 feet and breadth 5 feet.

Ans. 23.562 sq. ft.

MENSURATION OF VOLUMES.

757. A Volume is that which has length, breadth, and . thickness.

THE PRISM.

758. A Prism is a volume whose ends are equal polygons and whose sides are parallelograms.

759. The polygons are called bases, the paralelograms form the convex surface, and the prism takes its name from the form of its bases.

760. The Parallelopipedon is a prism whose bases are parallelograms. A cube is a parallelopipedon all of whose sides are squares.

761. Rule. To find the convex surface of a prism, multiply the perimeter of the base by the height.

NOTE.-To find the entire surface we add the area of the bases.

1. What is the convex surface of a triangular prism, the three sides of whose base are respectively 6, 7, and 8 inches, and height 50 inches? Ans. 1050 sq. in.

2. What is the entire surface of the triangular prism given Ans. 1090.66 sq. in.

in the first problem?

762. Rule.-To find the contents of a prism, multiply the area of the base by the altitude of the prism.

1. What are the contents of a square prism whose altitude is 30 feet, and the side of the base 3 feet? Ans. 270 cu. ft. 2. Required the contents of a triangular prism, the sides of whose base are each 16 inches, and whose altitude is 20 inches. Ans. 2217.02 cu. in.

THE PYRAMID.

763. The Pyramid is a volume bounded by a polygon and several triangles meeting in a common point. The polygon is called the base, and the triangles form the convex surface.

764. The point at the top is called the vertex, the distance from the vertex to the base is the

altitude, and from the vertex to the middle of a side is the slant height.

765. Rule.-To find the convex surface of a pyramid, multiply the perimeter of the base by one-half the slant height.

1. What is the convex surface of a triangular pyramid whose sides are each 4 ft. and slant height 27 ft.?

Ans. 162 sq. ft. 2. Required the convex surface of a pentangular pyramid whose sides are each 5 ft. and slant height 60 ft.

Ans. 750 sq. ft. 766. Rule.—To find the contents of a pyramid, multiply the area of the base by one-third of the altitude.

1. Required the contents of a pyramid whose base is 8 ft. square, and whose altitude is 69 ft. Ans. 1472 cu. ft.