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 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 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

B

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 in the first problem? Ans. 1090.66 sq. in.

1

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.

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 height

THE FRUSTUM OF A PYRAMID AND CONE. 411

1. What is the convex surface of a cone, the circumfer ence of whose base is 64 inches and slant height 40 inches? Ans. 1280 sq. in.

2. I have a conical haystack whose slant height is 8.25 ft., and the diameter of the base 6.5 ft.; how many square yards of canvas will cover it completely? Ans. 9.35935 sq. yd.

774. Rule. To find the contents of a cone, multiply the area of the base by one-third of the altitude.

1. Required the contents of a sugar-loaf, diameter of the base being 8 in. and height 18 in. Ans. 301.5936 cu. in. 2. How many cubic feet in a conical haystack 6 ft. high and 20 ft. in circumference? Ans. 63.664 cu. ft.

THE FRUSTUM OF A PYRAMID AND CONE.

775. The Frustum of a Pyramid is the part of a pyramid which remains after cutting off the top by a plane parallel to the base.

776. The Frustum of a Cone is the part of a cone which remains after cutting off the top by a plane parallel to the base.

777. Rule. To find the convex surface of a frustum, take the sum of the perimeters or circumferences of the two bases, and multiply it by one-half the slant height.

1. Required the convex surface of the frustum of a square pyramid whose slant height is 24 feet, the side of the lower base 12 feet, and upper base 8 feet. Ans. 960 sq. ft.

2. Required the surface of a frustum slant height is 20 feet, diameter of lower upper base 8 feet.

of a cone whose base 12 feet, and Ans. 628.32 sq. ft.

778. Rule. To find the contents of a frustum, take the sum of the two bases and the square root of their product, and multiply this sum by one-third of the altitude of the frustum.