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If, now, the number of sides of the regular polygon be continually doubled, the perimeter p decreases toward c as limit, and N toward circle. But the variables N and p are always in the constant ratio žr; therefore, by 798, Principle of Limits, their limits are in the same ratio,

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RULE. Multiply the length of the arc by half the radius.
FORMULA. S = {lr = kura.
PROOF. By 506, S:0::1:C:: U : 277,

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812. An Annulus is the figure included between two concentric circles. Its height is the difference between the radii.

813. To find the area of a sector of an annulus.

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Z 림 RULE. Multiply the sum of the bounding arcs by half the difference of their radii.

FORMULA. S. A. = 1 (r2 — r.) (ls + l) = Ihl, + 12).

PROOF. The annular sector is the difference between the two sectors fral, and Irali.

But I, and l, are arcs subtending the same angle; therefore, by 804,

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:: frakz - fril, = fralı + ralz - irili - fral

1(1– .)(1; + h). 814. To find the lateral area of a prism.

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RULE. Multiply a lateral edge by the perimeter of a right section.

FORMULA. P = lp.

PROOF. The lateral edges of a prism are all equal. The sides of a right section, being perpendicular to the lateral edges, are the altitudes of the parallelograms which form the lateral surface of the prism.

815. A Cylindric Surface is generated by a line so moving that every two of its positions are parallel.

816. The generatrix in any position is called an Element of the surface.

817. A Cylinder is a solid bounded by a cylindric surface and two parallel planes.

818. The Axis of a circular cylinder is the sect joining the centers of its bases.

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819. A Truncated Cylinder is the portion between the base and a non-parallel section.

820. To find the lateral area of a right circular cylinder.



RULE. Multiply its length by its circle.

PROOF. Imagine the curved surface slit along an element and then spread out flat. It thus becomes a rectangle, having for one side the circle, and for the adjacent side an element.

821. COROLLARY I. The curved surface of a truncated circular cylinder is the product of the circle of the cylinder by the intercepted axis.

For, by symmetry, substituting an oblique for the right section through the same point of the axis, alters neither the curved surface nor the volume, since the solid between the two sections will be the same above and below the right section.

822. COROLLARY II. The lateral area of any cylinder on any base equals the length of the cylinder multiplied by the perimeter of a right section.

823. The lateral surface of a prism or cylinder is called its Mantel.

EXERCISES. 114. The mantel of a right circular cylinder is equivalent to a circle whose radius is a mean proportional between the altitude of the cylinder and the diameter of its base.

115. A plane through two elements of a right circular cylinder cuts its base in a chord which subtends at the center an angle whose circular measure is u; find the ratio of the curved surfaces of the two parts of the cylinder.

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