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8. 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.

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 areą 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. Ans. 72 A. 56 P.-.

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THE CIRCLE.

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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 rd. 3. A man has a garden in the form of a circle, the diame ter 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 in 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 1.- 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 <ifferent 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.

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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 cat out of a circular board whose diameter is 14 inches ?

Ans. 9.899 in. 2. How large a square can be cat out of a circular board whose circumference is 200 inches? Ans. 45.0158 in.

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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 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.

NOTz.-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 in the first problem?

Ans. 1090.66 sq. in. 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, multvply the area of the base by one-third of the altitude.

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

2. Required the contents of a pyramid whose base is a triangle, each side of which is 8 ft., and the altitude of the pyramid 69 ft.

Ans. 637.376 cu. ft.

THE CYLINDER. 767. The Cylinder is a round body of uniform diameter with circles for its ends. The two circular ends are called bases.

768. The Altitude of a cylinder is the distance from the centre of one base to the centre of the other.

769. Rule.- To find the convex surface of a cylinder, multiply the circumference of the base by the altitude.

1. What is the convex surface of a cylinder, altitude 12 ft. and diameter of base 6 ft. ? Ans. 226.1952 sq. ft.

2. What is the convex surface of a cylinder 40 feet long and 15 feet in diameter ?

Ans. 1884.96 sq. ft. 770. Rule.- To find the contents of a cylinder, multiply the area of the base by the altitude.

1. Required the contents of a cylinder 60 feet long and 8 feet in diameter.

Ans. 3015.936 cu. ft. 2. Required the contents of a cylindrical log 12 feet long and 6 feet in diameter.

Ans. 418.88 cu. ft.

THE CONE. 771. A Cone is a volume whose base is a circle, and whose convex surface tapers uniformly to a point called the vertex.

772. The Altitude of a cone is the distance from the vertex to the centre of the base, and the slant height is the distance from the vertex to the circumference of the base.

773. Rule.- To find the convex surface of a cone, multiply the circumference of the base by one-half the slant herght

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