THEOREM VI.

The volume of any prism is equal to the product of its base by its altitude.

Let V denote the volume of any prism, B its base, and a its altitude; then

V=BX a.

For, let V' denote the volume of a triangular prism having

an equal altitude a', and an equal base T.

1. Any two prisms are proportional to the products of their bases by their altitudes.

2. Any two prisms having equal altitudes, are proportional to their bases.

3. Any two prisms having equal bases, are proportional to their altitudes.

Note. It is worthy of notice, that the volume of a prism depends conjointly on its altitude, and the area of its base;-with the same base, the volume will vary as the altitude; with the same altitude, the volume will vary as the base.

CHAPTER XIV.

THE PYRAMID.

DEFINITIONS.

1. A Pyramid is a polyedron, or volume, having a polygon for its base, and triangles for its lateral faces.

The vertex of the pyramid is the common vertex of the triangles; the lateral surface of the pyramid is the sum of its lateral faces; the lateral edges of the pyramid are the lines in which the lateral faces meet.

2. The Altitude of a pyramid is the perpendicular distance from its vertex to the plane of its base.

3. A Triangular pyramid is one whose base is a triangle.

4. A Right Pyramid is one whose base is a regular polygon, and whose perpendicular from the vertex pierces the centre of the base.

5. The Slant Height of a right pyramid is the altitude of any lateral face.

6. A Frustum of a pyramid is the portion of a pyramid included between the base and a parallel section.

If the cutting plane is not parallel to the base, the intercepted portion is called a Truncated pyramid.

7. The Altitude of a Frustum is the perpendicular distance between the planes of its bases.

THEOREM VI.

The volume of any prism is equal to the product of its base by its altitude.

Let V denote the volume of any prism, B its base, and a its altitude; then

V=BX a.

For, let V' denote the volume of a triangular prism having

an equal altitude a', and an equal base T.

1. Any two prisms are proportional to the products of their bases by their altitudes.

2. Any two prisms having equal altitudes, are proportional to their bases.

3. Any two prisms having equal bases, are proportional to their altitudes.

Note. It is worthy of notice, that the volume of a prism depends conjointly on its altitude, and the area of its base;-with the same base, the volume will vary as the altitude; with the same altitude, the volume will vary as the base.

CHAPTER XIV.

THE PYRAMID.

DEFINITIONS.

1. A Pyramid is a polyedron, or volume, having a polygon for its base, and triangles for its lateral faces.

The vertex of the pyramid is the common vertex of the triangles; the lateral surface of the pyramid is the sum of its lateral faces; the lateral edges of the pyramid are the lines in which the lateral faces meet.

2. The Altitude of a pyramid is the perpendicular distance from its vertex to the plane of its base.

3. A Triangular pyramid is one whose base is a triangle.

4. A Right Pyramid is one whose base is a regular polygon, and whose perpendicular from the vertex pierces the centre of the base.

5. The Slant Height of a right pyramid is the altitude of any lateral face.

6. A Frustum of a pyramid is the portion of a pyramid included between the base and a parallel section.

If the cutting plane is not parallel to the base, the intercepted portion is called a Truncated pyramid.

7. The Altitude of a Frustum is the perpendicular distance between the planes of its bases.

8. The Slant Height of a Frustum of a right pyramid is the altitude of any lateral face.

The lateral faces of such a frustum, as appears from the diagram, are trapezoids.

THEOREM I.

Any section of a pyramid parallel to the base is similar to the base.

In the pyramid S-ABCDE, let the section

a b c d e be parallel to the base; then will it be similar to the base.

For, all of its sides are parallel to the corresponding sides of the base. Ch. XII, Th. V. Hence, the base and the section are mutually equiangular. -Ch. XII, Th. VIII. Since ab and b c are parallel to A B and B C, the triangles S a b and SA B are similar

...

ab AB be: BC: ed: CD, etc. Hence, the section and the base have their homologous sides proportional.

Hence, they are similar.

Q. E. D.

THEOREM II.

If two pyramids have the same altitude, sections parallel to the bases and equally distant from them, are proportional to the bases.

Let S - ABCDE and R F G H be two pyramids, whose bases are in the same plane, and in which abcde and fgh are