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620. COROLLARY 1. The lateral area of a right cylinder is equal to the product of the perimeter of its base by its altitude.

621. COR. 2. Let a cylinder of revolution be generated by the rectangle whose sides are R and H revolving about the side H.

Then R is the radius of the base of the cylinder, and H the altitude of the cylinder.

hence,

The perimeter of the base is 2 R;
S= 2 TRX H.

The area of each base is π R2 ;

§ 381

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hence, the total area T' of a cylinder of revolution is expressed by

RX H2 π R2 π

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2 π R (H+ R).

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T: = 2 622. COR. 3. Let S, S' denote the lateral areas of two similar cylinders of revolution;

T, T' their total areas; R, R' the radii of their bases; H, H' their altitudes.

Since the generating rectangles are similar, we have

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That is, the lateral areas, or the total areas, of similar cylin

ders of revolution are to each other as the squares of their altitudes, or as the squares of the radii of their bases.

PROPOSITION XXX. THEOREM.

623. The volume of a cylinder is equal to the product of its base by its altitude.

B

G

Let V denote the volume of the cylinder A G, B its base, and H its altitude.

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Let V' denote the volume of the inscribed prism A G, B' its base, and H will be its altitude.

Then

V' = B' X H,

§ 543

(the volume of a prism is equal to the product of its base by its altitude). Now, let the number of lateral faces of the inscribed prism be indefinitely increased, the new edges continually bisecting the arcs of the bases.

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But however great the number of the lateral faces,

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624. COROLLARY 1. Let V be the volume of a cylinder of revolution, R the radius of its base, and H its altitude.

Then the area of its base is π R2,

§ 381

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625. COR. 2. Let V and V' be the volumes of two similar cylinders of revolution, R and R' the radii of their bases, H and H their altitudes.

Since the generating rectangles are similar, we have

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That is, the volumes of similar cylinders of revolution are to each other as the cubes of their altitudes, or as the cubes of the radii of their bases.

Ex. 1. Required, the entire surface and volume of a cylinder of revolution whose altitude is 30 inches, and whose base is a circle of which the diameter is 20 inches.

2. Required, the volume of a right truncated triangular prism the area of whose base is 40 inches, and whose lateral edges are 10, 12, and 15 inches, respectively.

3. Let E denote an edge of a regular tetrahedron; show that the altitude of the tetrahedron is equal to EV; that the surface is equal to E2√3; and that the volume is equal to

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4. Required, the number of quarts that a cylinder of revolution will contain whose height is 20 inches, and whose diameter is 12 inches.

5. Given S, the surface of a cube, find its edge, diagonal, and volume. What do these become when S54?

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626. Through a given point to pass a plane tangent to a given cylinder.

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CASE I. When the given point is in the curved surface of the cylinder.

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Let A C' be a given cylinder, and let the given point be a point in the element A A'.

It is required to pass a plane tangent to the cylinder and embracing the element A A'.

Draw the radius O A, and A T tangent to the base;

and pass a plane RT" through A A' and A T.

The plane RT" is the plane required.

For, through any point P in this plane, not in the element A A',

pass a plane to the base, intersecting the cylinder in the MN,

and the plane RT' in MP.

From the centre of the OMN draw QM.

MP and MQ are Il respectively to A T and AO,
(the intersections of two || planes by a third plane are || lines);

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§ 462

(two not in the same plane, having their sides || and lying in the same

direction, are equal).

.. PM is tangent to the O M N at M.

.. P lies without the O M N,

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to it.

and hence without the cylinder.

.. the plane R T' does not cut the cylinder, and is tangent

CASE II.

When the given point is without the cylinder.

Let P be the given point.

It is required to pass a plane through P tangent to the cylinder.

Through P draw the line PT to the elements of the cylinder,

meeting the plane of the base at T.

From 7 draw TA and TC tangents to the base. § 240 Through P T and the tangent TA pass a plane R T'.

Since

A A' is to PT,

Cons.

the plane R T', passing through PT and the point A will contain the element A A',

(two || lines lie in the same plane).

And, since R T also contains the tangent A T,

it is a tangent plane to the cylinder.

In like manner, the plane T S', passed through PT and the tangent line TC,

is a tangent plane to the cylinder.

Q. E. F.

627. COROLLARY 1. The intersection of two tangent planes to a cylinder is parallel to the elements of the cylinder.

628. COR. 2. Any straight line drawn in a tangent plane, and cutting the element of contact, is tangent to the cylinder.

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