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THEOREM VIII.

The convex surface of a cylinder is equal to the circumference of its base multiplied by its altitude; and its solidity is equal to the area of its base multiplied by its altitude.

Let ABCD be a cylinder, having A a prism inscribed in it whose base is a regular polygon. Now, if the number of sides of this polygon be indefinitely increased, its perimeter will ultimately coincide with the D circumference of the base of the

cylinder. Then, also, the convex surface of the prism will coincide with the convex surface of the cylinder, and the solidity of the prism with the solidity of the cylinder. But the convex surface of the prism is equal to the perimeter of the base multiplied by the altitude (Theo. VI), and its solidity is equal to the area of the base multiplied by the altitude (Theo. VII).

Therefore, the convex surface of a cylinder, etc.

SEC. XVI.-PYRAMIDS AND CONES.

DEFINITIONS.

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1. A PYRAMID is a solid bounded by plane faces, of which one is any polygon, and the others triangles having a common vertex. The polygon is called the base. The triangles together form the convex or lateral surface.

Pyramids are called triangular, quadrangular, pen

tagonal, etc., according as their bases are triangles, quadrilaterals, pentagons, etc.

2. A regular PYRAMID is one whose base is a regular polygon, and the triangular faces are equal and isosceles.

3. A CONE is a solid described by the revolution of a right-angled triangle about one of

the sides containing the right angle, which side remains fixed. The fixed side is called the axis of the cone. The hypotenuse describes the convex surface. The circle described by the other revolving side is called the base.

4. The ALTITUDE of a pyramid or cone is the perpendicular distance from the vertex to the plane of the base.

5. The slant hight of a regular pyramid is the perpendicular let fall from the vertex upon the base of any one of its triangular faces. The side or slant hight of a cone is the straight line drawn from the vertex to any point in the circumference of the base.

6. A FRUSTUM of a pyramid or a cone, is the portion next the base cut off by a plane parallel to the base. The slant hight of a frustum is that part of the slant hight of the whole solid which lies on the frustum.

7. A SECTION of any solid is the surface in which it is divided by a plane which passes through it.

THEOREM IX.

The convex surface of a regular pyramid is equal to half the product of the perimeter of the base by the slant hight.

B

1

E

Let ABCDE be a regular pyramid, of which AF is the slant hight. The area of the triangle ACD is equal to half the product of its base CD into its altitude, which is the slant hight. Consequently the areas of all the equal triangles composing the convex surface are together equal to half the product of the sum of their bases by the slant hight. But the sum of their bases constitutes the perimeter of the base of the pyramid. Therefore, the convex surface of a regular pyramid, etc.

Schol. 1. The trapezoids BF, CG, etc., composing the convex surface of a frustum of a regular pyramid, are equal to each other; for they are the differences between the equal triangles AEF, AFG, etc., and the equal triangles ABC, ACD, etc.

E

A

BAR

D

L

N

G

Schol. 2. Since the area of the trapezoid BF is equal (Cor. 2, Theo. IX, B. II) to its mean breadth LM multiplied by RS, which is the slant hight of the frustum, and since the same is true of each of the other trapezoids, it follows that the convex surface of a frustum of a regular pyramid is equal to its slant

hight multiplied into the perimeter of a middle section between its two bases.

THEOREM X.

Two triangular pyramids of equal bases and altitudes are equivalent.

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and LMN. Now, since the vertices R and S are by hypothesis equidistant above the plane of the bases, a third plane may pass through these two points parallel with the other two planes (Cor. 2, Theo. II). Then the lines HL and RS, being the intersections of two parallel planes by a third plane CRS, will be parallel (Def. 4, Sec. XIII); hence, the triangles CRS, CHL, are similar (Cor., Theo. V, B. II). In the same manner it may be shown that the triangles CRB, HRG, are similar; also, CSD and LSM.

Now, CR HR: CB HG.

Also, CS LS:: CD: LM.

But by similarity of CRS and CHL, we have
CR HR CS: LS.

:

Hence, by equality of ratios (Def. 9, Sec. X),

CB: HG: CD: LM.

But CB is by hypothesis equal to CD; therefore, HG is equal to LM. In the same way it may be shown Evans' Geometry.—8

that the other sides of the triangle FHG are respectively equal to the other sides of the triangle LMN; and these triangles are consequently equal throughout. Hence, at equal hights, the sections parallel to the bases are equal; and the two pyramids may be conceived to be applied to one another so as to coincide at all equal hights successively, from their bases to their vertices. They are, therefore, equivalent.

That is, two triangular pyramids, etc.

THEOREM XI.

A triangular pyramid is one-third of a triangular prism of the same base and altitude.

Let ABCDEF be a triangular prism. Join AF, BF, and BD. Now, the pyramid BDEF, cut off by the plane of the triangle BDF, is equivalent (Theo. X) to the pyramid FABC, cut off by the plane of the triangle ABF; for they have equal bases, DEF and ABC (Def. 1, Sec. XV), and the

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same altitude, namely the altitude of the prism. But the pyramid FABC is equivalent to the third pyramid BADF; for they have equal bases ADF and FCA (Cor. 1, Theo. XIII, B. I), and the same altitude, namely the perpendicular distance of their common vertex B above the plane of their bases ADFC. Hence, the pyramid BDEF is one-third of the prism ABCDEF.

That is, a triangular pyramid is one-third, etc.

Cor. 1. Hence, the solidity of a triangular pyramid

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