Of Solids bounded by Planes.

SECTION X.

OF SOLIDS BOUNDED BY PLANES.

1. A solid or body was defined (§ I. Art. 8), to be that which has length, breadth, and thickness.

2. Every solid bounded by planes is called a polyedron.

3. The planes which bound a polyedron are called faces. The straight lines in which the faces intersect each other, are called the edges of the polyedron; and the points at which the edges intersect, are called the vertices of the angles, or vertices of the polyedron.

QUEST.-1. What is a solid ? 2. What is a polyedron ? 3. What are the planes called which bound a polyedron? What are the lines called in which the faces intersect each other? What are the points called in which the edges intersect each other?

Of Solids bounded by Planes.

4. A prism is a solid, whose ends are equal polygons, and whose side faces are parallelograms.

Thus, the prism whose lower base is the pentagon ABCDE, terminates in an equal and parallel pentagon FGHIK, which is called the upper base. The side

F

A

B

C

faces of the prism are the parallelograms_ DH, DK, EF, AG and BH. These are called the convex or lateral surface of the prism.

5. The altitude of a prism is the distance between its upper and lower bases: that is, it is a line drawn from a point of the upper base, perpendicular to the lower base. 6. A right prism is one in which the edges AF, BG, EK, F HC, and DI are perpendicular to the bases. In the right prism,

either of the perpendicular edges

is equal to the altitude. In the oblique prism the altitude is less than the edge.

B

E

QUEST.-4. What is a prism? What is the lower base of a prism? What is the upper base? What are the side faces of the prism, taken together, called? 5. What is the altitude of a prism? 6. What is a right prism? In a right prism what is one of the perpendicular edges equal to ? In the oblique prism is the edge greater or less than the altitude?

Of Solids bounded by Planes.

7. A prism whose base is a triangle, is called a triangular prism: if the base is a quadrangle it is called a quadrangular prism: if a pentagon, a pentagonal prism: if a hexagon it is called a hexagonal prism, &c.

8. A prism whose base is a parallelogram, and all of whose faces are also parallelograms, is called a parallelopipedon. If all the faces are rectangles, it is called a rectangular parallelopipedon. If all the faces are squares it is called a cube. The cube is bounded by six equal faces at right angles to each other.

9. The opposite faces of a parallelopipedon are equal to each other. Thus, the parallelogram BD is equal to the opposite parallelogram FH, the parallelogram BE to CH, and BG to AH.

QUEST.-7. What is a triangular prism? What is a quadrangular prism? What is a pentagonal prism? 8. What is a parallelopipedon? When all the faces are rectangles, what is it called? If all the faces are squares, what is it then called? How is a cube bounded? 9. Are the opposite faces of a parallelopipedon equal or unequal? Point out the equal faces?

Of Solids bounded by Planes.

10. A pyramid is a solid, formed by several triangles united at the same point S, and terminating in the different sides of a polygon ABCDE.

E

The polygon ABCDE, is called the base of the pyramid; the point S, is called the vertex, and the triangles ASB, BSC, CSD, DSE, and ESA, form its lateral, or convex surface.

S

A

B

11. The altitude of a pyramid, is the perpendicular let fall from the vertex, upon the plane of the base.

Thus, SO is the altitude of the pyramid S-ABCDE.

A

E

B

QUEST.-10. What is a pyramid? What is the base of a pyramid ? Point it out. What is the vertex of a pyramid? What are the triangles called, which bound the pyramid ? 11. What is the altitude of a pyramid ? Point out the altitude.

Of Solids bounded by Planes.

12. The slant height of a regular pyramid, is a line drawn from the vertex, perpendicular to one of the sides of the polygon which forms its base. Thus, SF is the slant height of the pyramid S— ABCDE.

E

B

F

13. When the base of the pyramid is a regular polygon, and the perpendicular SO, passes through the middle point of the base, the pyramid is called a regular pyramid, and the line SO, is called the axis.

14. If from the pyramid SABCDE, the pyramid S-abcde be cut off by a plane parallel to the base, the remaining solid, below the plane, is called the frustum of a pyramid.

The altitude of a frustum is the perpendicular distance between the upper and lower planes.

Ε

a

b

B

QUEST.-12. What is the slant height of a regular pyramid? Point out the slant height. 13. When is a pyramid regular? In a regular pyramid, what is the axis? 14. What is the frustum of a pyramid? What is the altitude of a frustum ?