The slant height of any side of a frustrum of a pyramid is measured from the middle points of the top and bottom sides of the trapezium forming that side.

To find the surface of any frustrum of a pyramid, take the sum of the areas of all the trapeziums forming the sides, to which add the sum of the top and base.

The surface of a frustrum of a right regular pyramid, where the top and base are parallel planes, is equal to one-half the sum of the perimeters of top and base multiplied by the slant height plus the sum of the areas of the top and base.

The volume of any frustrum of any pyramid, with top and base parallel, is equal to one-third the perpendicular distance between top and base multiplied by the sums of the areas of top and base, and the square root of the product of those areas.

Thus H, being the perpendicular and A and A' the areas of top and base, respectively, then the volume equals H x (A + A' + √Ā × A') or A" being equal to the area of a section midway between and parallel to base and top, the volume V = { H (A + A' + 4 A′′).

=

A prismoid is a solid having six sides, two of which are parallel but unequal quadrangles, and the other sides trapeziums.

To find the Volume of a Prismoid

Let A = area of one of the parallel sides.

Then

a = area of the other parallel side.

M

= area of cross section midway between and parallel to the parallel sides.

L = perpendicular distance between the two parallel sides.

The wedge is a frustrum of a triangular prism. Its volume is equal to the area of a right section multiplied by one-third the sum of the lengths of the three parallel edges.

Let A equal area of section perpendicular to the axis of the prism and BC, DE and FG, the lengths of the parallel edges respectively.

A polyhedron is a solid bounded by plane surfaces.

A regular polyhedron is one whose bounding faces are all equal and regular polygons.

A cylinder may be defined as a prism, of which a section perpendicular to its axis is a circle. It may be right or oblique.

The base of a right cylinder is a circle, that of an oblique cylinder an ellipse.

The surface of any cylinder is equal to the product of the circumference of a circle whose plane is perpendicular to the axis of the cylinder, by the length of the axis, plus the area of the ends.

The volume of a cylinder is equal to the area of a circle perpendicular to the axis multiplied by its altitude.

The Cone

A cone is a pyramid having an infinite number of sides.

Cones are right or oblique according as their axes are perpendicular or inclined to their bases.

The surface of a right cone is equal to the product of the perimeter of the base by half the slant height, plus the area of the base.

The surface of an oblique cone, cut from a right cone having a circular base, is equal to the area of the base, multiplied by the altitude and divided by the perpendicular distance from the axis at the point where it pierces the base, to the surface of the cone, plus the area of the base; AH or the curved surface of the cone equals Wherein A is the area R

of the base, H the altitude and R the perpendicular.

The volume of any cone is equal to the area of the base multiplied by one-third of the altitude.

The volume of a cone is equal to one-third that of a cylinder, or onehalf that of an hemisphere having same base and altitude.

The surface of a right circular frustrum of a cone with top and base parallel is found by adding together the circumferences of top and base, multiplying this sum by one-half the slant height; to this product add the area of top and base to get the total surface.

The volume of a frustrum of any cone, with top and base parallel, is equal to one-third of the altitude multiplied by the sum of the areas of top and base plus the square root of the product of those areas, or equals the altitude

× (area of top + area of base + √area of top × area of base).

The Sphere

A sphere is a solid generated by revolving a semicircle about its diameter.

The intersection of a sphere with any plane is a circle.

A circle cut by the intersection of the surface of a sphere and a plane passing through its center is a great circle.

The volume of a sphere is greater than that of any other solid having an equal surface.

The surface of a sphere equals that of four great circles.

=

4 πr2.

= πD2.

= curved surface of a circumscribing cylinder.

= area of a circle having twice the diameter of the sphere. The surface of a sphere is equal to that of a circumscribing cube multiplied by 0.5236.

Surfaces of spheres are to each other as the squares of their diameters.

=

Volume of a Sphere

= 4.18883

πD3 = 0.5236 D3

volume of circumscribing cylinder.

= 0.5236 volume of circumscribing cube.

Volumes of spheres are to each other as the cubes of their diameters.

The area of the curved surface of a spherical segment is equal to the product of the circumference of a great circle by the height of the segment = πDH, where D is the diameter of the sphere and H the height of the spherical segment.

The curved surface of a segment of a sphere is to the whole surface of the sphere as the height of the segment is to the diameter of the sphere.

To Find the Volume of a Spherical Segment

To find the curved surface of a spherical zone, multiply the circumference of the sphere by the height of the zone.

To find the volume of a spherical zone, let A and A' be the radii of the ends of the zone and H be the height and V the volume.

Then

V = } πH (3(A2 + A2') + H2).

Guldin's Theorems

(1) If any plane curve be revolved about any external axis situated in its plane, the surface generated is equal to the product of the perimeter of the curve and the length of the path described by the center of gravity of that perimeter.

(2) If any plane surface be revolved about any external axis situated in its plane, the volume generated is equal to the area of the revolving surface multiplied by the path described by its center of gravity.

CHAPTER II

WEIGHTS AND MEASURES

In the United States and Great Britain measures of length and weight are, for the same denomination, essentially equal; but liquid and dry measures for same denomination differ widely. The standard measure of length for both countries is that of the simple seconds pendulum, at the sea level, in the latitude of Greenwich; in vacuum and at a temperature of 62° F.

The length of such a pendulum is 39.1393 inches; 36 of these inches constitute the standard British Imperial yard. This is also the standard in the United States.

The Troy pound at the U. S. Mint of Philadelphia is the legal standard of weight in the United States.

It contains 5760 grains and is exactly the same as the Imperial Troy pound of Great Britain.

The avoirdupois pound (commercial) of the United States contains 7000 grains, and agrees with the British avoirdupois pound within 0.001 of a grain.

The metric system was legalized by the United States in 1866 but its use is not obligatory.

The metre is the unit of the metric system of lengths and was supposed to be one ten millionth, of that portion of a meridian between

I

10,000,000

either pole and the equator.

The metric measures of surface and volume are the squares and cubes of the metre, and of its decimal fractions and multiples.

The metric unit of weight is the gramme or grain, which is the weight

of a cubic centimeter of pure water at a temperature of 40° F.

The legal equivalent of the metre as established by Act of Congress is 39.37 inches = 3.28083 ft. 1.093611 yards.

=

The legal equivalent of the gramme is 15.432 grains.

The systems of weights used for commercial purposes in the United States are as follows: