tance between the ends is 13 feet, and the sides of the base 23, 34, and 19 inches. Ans. 85.22+ square feet. 2. Required the surface of a pentagonal prism, whose length is 14 feet, and each side of whose base is 33 inches. Ans. 218.52 sq. ft. 3. Required the surface of a cylinder 13 feet long, the circumference of whose base is 57 inches. Ans. 65.34 square feet.

4. How often must a cylinder, 5 feet 3 inches long, whose diameter is 21 inches, revolve, to roll an acre?

5. Required the wall-surface of a square 16 feet long and 10 feet high.

Ans. 1509.18 times. room, whose sides are each Ans. 71 square yards.

651. To find the contents or volume of a prism or cylinder. Multiply the area of the base of the given prism or cylinder by the height.

Ex. 1. What are the contents of a triangular prism, whose length is 12 feet, and each side of whose base is 24 feet? Ans. 32.47 cu. ft. 2. Required the volume of a triangular prism, whose length is 10 feet, and the three sides of whose triangular end or base are 5, 4, and 3 feet. Ans. 60 cu. ft.

3. How many cubic feet in a block of marble, whose length is 3 feet 2 inches, breadth 2 feet 8 inches, and depth 2 feet 6 inches? Ans. 21 cu. ft. 4. What is the volume of a cylinder, whose length is 9 feet, and the circumference of whose base is 6 feet? Ans. 25.78

PYRAMIDS AND CONES.

652. A pyramid is a solid having for its base some rectilineal figure, and for its sides triangles meeting in a common point called the vertex; as the figure A B.

The slant height of a pyramid is a line drawn from the vertex to the middle of one of the sides of the base.

cu. ft.

A

B

653. A cone is a solid having a circle for its base, and tapering uniformly to a point called the vertex.

The slant height of a cone is a line drawn from the vertex to the circumference of the base.

654. The altitude or height of a pyramid or of a cone is a line drawn from the vertex perpendicular to the plane of the base.

655. The frustum of a solid is the part that remains after cutting off the top by a plane parallel to the base; as the frustum of a cone C D.

Ꭰ

656. To find the surface of a pyramid or of a cone.

C

Multiply the perimeter or the circumference of the base by half of the

slant height, and to the product add the area of the base.

Ex. 1. Required the area of the surface of a square pyramid, whose base is 2 feet 8 inches square, and whose slant height is 3 feet 9 inches. 2. What is the convex surface of a cone, whose slant height is 20 feet, and the circumference of whose base is 9 feet? Ans. 90 feet.

657. To find the volume of a pyramid, or of a cone. Multiply the area of the base by one third of the altitude. Ex. 1. What is the solidity of a cone, whose height is 12 feet, and the diameter of whose base is 24 feet? Ans. 20.45+ feet. 2. What are the contents of a triangular pyramid, whose height is 14 feet 6 inches, and the sides of whose base are 5, 6, and 7 feet? Ans. 71.035 feet.

658. To find the surface of a frustum of a pyramid, or of a cone. Multiply the sum of the perimeters or of the circumferences of the two ends by half of the slant height; and to the product add the areas of the

two ends.

Ex. 1. Required the surface of a frustum of a pentagonal pyramid, whose slant height is 10 inches, and the sides of whose base are 3 and 5 inches.

2. What is the surface of the frustum of a cone, the diameters of the bases being 43 and 23 inches, and the slant height 9 feet? ft.

Ans. 90.72+ sq.

659. To find the volume of a frustum of a pyramid, or of a cone.

Multiply the areas of the two ends together, and extract the square root of the product. To this root add the two areas, and multiply their sum by one third of the altitude.

Ex. 1. If the length of a frustum of a square pyramid be 18 feet 8 inches, the side of its greater base 27 inches, and that of its less 16 inches, what is the volume ? Ans. 61.228+ cu. ft.

2. What are the contents of a stick of timber, whose length is 40 feet, the diameter of the larger end being 24 inches, and of the smaller end 12 inches? Ans. 73 cu. ft., nearly.

SPHERES, SPHEROIDS, &C.

660. A sphere is a solid, bounded by a curved surface, every part of which is equally distant from a point within, called the centre.

The axis or diameter of a sphere is a line passing through the centre, and terminated by the surface; as the line A B.

A

B

The radius of a sphere is a line drawn from the centre to any part of the surface.

661. A segment of a sphere is a part of it cut off by any plane; as the figure Á C B D. The plane is the base of the segment; the perpendicular distance from the center of the base to the convex surface is the height of the segment; as C D.

662. A spherical zone is a part of the surface of a sphere included between two parallel planes, which form its bases; and the height of a spherical zone is the perpendicular distance between the planes forming its bases.

663. A cylindrical ring is a figure formed by bending a cylinder uniformly till the two ends meet; as A CD B.

664. A spheroid is a figure resembling a sphere, and which may be formed by the revolution of an ellipse about one of its axes; as AECBDFA.

If the ellipse revolves about its longer or transverse axis or diameter, the spheroid is prolate, or oblong; if about its shorter or conjugate diameter, the spheroid is oblate, or flattened.

A

665. A segment of a spheroid is a part cut off by any plane, as FAEF.

666. To find the surface of a sphere.

Multiply the diameter by the circumference.

Ex. 1. Required the convex surface of a globe, whose diameter is

24 inches.

Ans. 1809.55+ inches. 2. Required the surface of the earth, its diameter being 7957 miles, and its circumference 25,000 miles. Ans. 198943750 square miles.

667. To find the solidity of a sphere.

Multiply the cube of the diameter by .523598.

Ex. 1. What is the solidity of a sphere, whose diameter is 12 inches? Ans. 904.78+ inches. 2. Required the solidity of the earth, supposing its circumference to be 25,000 miles. Ans. 263858149120.06886875 miles.

668. To find the convex surface of a segment or of a zone of a sphere.

Multiply the height of the segment or zone by the circumference of the sphere of which it is a part.

Ex. 1. If the diameter of a sphere is 12 feet, what is the convex surface of a segment cut off from it, whose height is 2 feet?

Ans. 78.54 sq. ft.

2. If the diameter of the earth, considered as a perfect sphere, is 7970 miles, and the height of each temperate zone is taken at 2143.623553 miles, what is the surface of each temperate zone? Ans. 53673093.12+ sq. m.

669. To find the solidity of a segment of a sphere.

Multiply the square of the height plus three times the square of the radius of the base, by the height, and this product by .5236.

Ex. 1. Required the solidity of a spherical segment, whose height is 3 feet, and the radius of whose base is 4 feet. Ans. 109.56 cu. ft. 2. Required the solidity of the segment of a sphere, whose height is 9 feet, and the diameter of whose base is 20 feet.

Ans. 1795.42 cu. ft.

670. To find the surface of a cylindrical ring.

Multiply the sum of the thickness and the inner diameter by the thickness, and that product by 9.8696.

Ex. 1. Required the surface of a cylindrical ring, whose inner diameter is 21 inches, and whose thickness is 4 inches.

Ans. 986.96 sq. in.

671. To find the solidity of a cylindrical ring.

Multiply the sum of the thickness and the inner diameter by the square of the thickness, and that product by 2.4674.

Ex. 1. Required the solidity of a cylindrical ring, whose inner diameter is 25 inches, and whose thickness is 5 inches.

672. To find the solidity of a spheroid.

Ans. 1850.55 cu. in.

Multiply the square of the revolving axis by the fixed axis, and that product by .523598.

Ex. 1. If the fixed axis of a spheroid is 32 inches, and the revolving axis 20 inches, what is the solidity? Ans. 6702.08 cu. in.

2. Required the contents of a balloon in the form of a prolate spheroid, having its longer diameter 48 feet, and its shorter 38 feet. Ans. 36291.76 cu. ft.

MENSURATION OF LUMBER.

673. Boards are usually measured by the square foot.

Planks, joists, beams, &c. are usually measured by board measure, the board being considered to be 1 inch in thickness.

Round timber is sometimes measured by the ton, and sometimes by board measure.

674. To find the number of square feet in a board.

Multiply the length of the board, taken in feet, by its width, taken in inches, and the product divided by 12 will give the contents in square feet Or,

Take both the length and width in feet, and their product will be the contents in feet

NOTE. If the board is tapering, take half the sum of the width of its ends for the width.

Ex. 1. What are the contents of a board 24 feet long, and 8 inches wide? Ans. 16 feet. 2. What are the contents of a board 30 feet long, and 16 inches wide ? Ans. 40 feet. 3. What are the contents of a tapering board, 30 feet long, whose ends are, the one 26 inches, and the other 14 inches wide?

675. To find the number of feet, board measure, in a plank, joist, beam, &c.

Multiply the width taken in inches by the thickness in inches, and this product by the length, in feet; and the last product divided by 12 will give the contents in feet, board measure.

NOTE. If the plank, joist, &c. is tapering in width, take half the sum of the width of the ends for the width; and if the taper be both of the width and the thickness, the common rule of obtaining the contents in cubic feet is, to multiply half the sum of the areas of the two ends by the length, and divide the product by 144.

Ex. 1. How many feet are there in 3 joists, which are 15 feet long, 5 inches wide, and 3 inches thick? Ans. 564 feet. 2. How many feet in 20 joists, 10 feet long, 6 inches wide, and 2 inches thick ? Ans. 200 feet. 3. How many feet in a beam 20 feet long, 10 inches thick, whose width tapers from 18 to 16 inches? Ans. 283 feet.

676. To find the contents of round timber.

Multiply the length, taken in feet, by the square of one fourth of the mean girth, taken in inches, and this product divided by 144 will give the contents in cubic feet.

NOTE. The girth of tapering timber is usually taken about the distance from the larger to the smaller end.

The rule is that in common use, though very far from giving the actual number of cubic feet in round lumber measured by it. 40 cubic feet, as given by the rule, are in fact equal to 50% true cubic feet. The following rule gives results more nearly accurate, requiring to be diminished by only one foot in 190, to give exact contents. Multiply the square of one fifth of the mean girth, taken in inches, by twice the length, in feet; and divide by 144.

Ex. 1. How many cubic feet in a stick of timber which is 30 feet long, and whose girth is 40 inches? Ans. 20 feet.

feet.

2. If a stick of timber is 50 feet long, and its girth is 56 inches, what number of cubic feet does it contain? 3. What are the contents of a log 90 feet long, ference is 120 inches?

Ans. 68 and whose circumAns. 562 feet.