of an equivalent right circular cylinder whose base is equal in area to the section of the frustum made by a plane parallel to its bases, and equidistant from the bases.

V = }πH (R2 + r2 + Rr)

§ 730

§ 700

Ex. 721. Find the edge of a cube equivalent to a regular tetrahedron whose edge measures 3 inches.

Altitude of base = √9} = √ √3.

of √3 = √3.

§ 372

Ex. 722. Find the edge of a cube equivalent to a regular octahedron whose edge measures 3 inches.

Let H denote the altitude of each of the two pyramids into which the octahedron can be divided, and V the volume of each pyramid.

Ex. 723. The dimensions of a trunk are 4 feet, 3 feet, 2 feet. Find the dimensions of a trunk similar in shape that will hold four times as much.

Ex. 724. By what number must the dimensions of a cylinder be multiplied to obtain a similar cylinder (i) whose surface shall be n times that of the first; (ii) whose volume shall be n times that of the first?

Ex. 725. A pyramid is cut by a plane parallel to the base which passes midway between the vertex and the plane of the base. Compare the volumes of the entire pyramid and the pyramid cut off.

Ex. 726. The height of a regular hexagonal pyramid is 36 feet, and one side of the base is 6 feet. What are the dimensions of a similar pyramid

Ex. 727. The length of one of the lateral edges of a pyramid is 4 meters. How far from the vertex will this edge be cut by a plane parallel to the base, which divides the pyramid into two equivalent parts?

Ex. 728. A lateral edge of a pyramid is a. At what distances from the vertex will this edge be cut by two planes parallel to the base that divide the pyramid into three equivalent parts?

Ex. 729. A lateral edge of a pyramid is a. At what distance from the vertex will this edge be cut by a plane parallel to the base that divides the pyramid into two parts which are to each other as 3:4 ?

Ex. 730. The volumes of two similar cones are 54 cubic feet and 432 cubic feet. The height of the first is 6 feet; what is the height of the other?

Ex. 731. Two right circular cylinders have their diameters equal to their heights. Their volumes are as 3: 4. Find the ratio of their heights.

Ex. 732. Find the dimensions of a right circular cylinder 15 as large as a similar cylinder whose height is 20 feet, and diameter 10 feet.

Ex. 733.

base is R.

: 1:0.97871.

= 10 x 0.97871 ft.
= 9.7871 ft.

The height of a cone of revolution is H, and the radius of its
Find the dimensions of a similar cone three times as large.
H: H' = VV: VV'. § 726

H: H' =1:

3

V/3.

R: RVV:VV.

R' =

§ 726

10:2 R'

=

.. 2 R'

.. H′ = H √3.

R√3.

Ex. 734. The height of the frustum of a right cone is the height of the entire cone. Compare the volumes of the frustum and the cone.

Ex. 735. The frustum of a pyramid is 8 feet high, and two homologous edges of its bases are 4 feet and 3 feet, respectively.

of the frustum and that of the entire pyramid.

Compare the volume

§ 672

BOOK VIII. SOLID GEOMETRY.

Ex. 736. Every point in a great circle which bisects a given arc of a great circle at right angles is equidistant from the extremities of the given

arc.

Let AC be any given arc of a great O, and let PBP' be the great which bisects the arc AC at right angles.

To prove that every point in the great O PBP' is equidistant from A and C.

Proof. Draw the chords AB, AC, BC.
Chord AB = chord BC.

P

A

B C

§ 241

○ PBP' passes through the poles of arc AC. § 754 But the poles of arc AC are equidistant from A and C, the distances being measured on straight lines.

Hence, the plane of the © PBP′ is 1 to the chord AC at its middle point. §§ 496, 517

Therefore, every point in the great ○ PBP' is equidistant from the points A and C.

Q. E. D.

Ex. 737. The radius of a sphere is 4 inches. From any point on the surface as a pole a circle is described upon the sphere with an opening of the compasses equal to 3 inches. Find the area of this circle.

Let O be the centre of the sphere, P the pole, and A a point in the O. In the plane ▲ POA, PO = OA = 4, and PA = 3.

is

Find the radii R, R'

Let D-ABC be the given regular tetrahedron whose edge is a.

The radius of the O circumscribed about each face of the tetrahedron

Let M be the centre of the O circumscribed about the ▲ ACD, and M' be the centre of the O circumscribed about the AABC. Is MO and M'O to the faces ACD and ABC, respectively.

Erect the

Ex. 739. Find the diameter of the section of a sphere 10 inches in diameter made by a plane 3 inches from the centre.

Let r denote the radius and d the diameter of the O made by the intersecting plane.

§ 743

The plane formed by drawing the radii of the sphere to the extremities of a diameter of the section is an isoscles ▲ with legs each 5 in. long, and with altitude 3 in.

Ex. 740. At a given point in a given arc of a great circle, to construct

a spherical angle equal to a given spherical angle.

Let A' be any point in the arc ED, and BAC any given Z.
To construct an equal to BAC having

A' as its vertex, and A'D as one side.

Construction. With A as pole and a

quadrant as radius arc, draw the arc BC.

With A' as a pole, draw the arc of a

great O B'm.

B

m

On this arc lay off B'C' equal to BC, and through A' and C' draw the arc of a great O A'C'. Then B'A'C' is the

required.

Proof. The arcs BC and B'C' are the measures of the A and A',

Another B'A'C" may be constructed below the line A'D.

Q. E. F.