VOLUMES OF PYRAMIDS AND CONES 730. Inscribed Prisms. If the altitude of a triangular pyramid is divided into equal parts by a series of planes parallel to the base, the prisms within the pyramid and having the sections formed by the parallel planes as bases, and each having one of the segments of one of the lateral edges of the pyramid as one of its lateral edges, are called a set of inscribed prisms. In the figure M, N, and R are a set of inscribed prisms, each being entirely within the pyramid. 731. Circumscribed Prisms. In a similar manner, the prisms having the base of the pyramid and the sections as bases are called a set of circumscribed prisms. In the figure, S, T, U, and V are a set of circumscribed prisms, each being partly outside of the pyramid. 732. The two sets of such prisms, formed by using the same lateral edge and the same parallel planes, are called corresponding sets of inscribed and circumscribed prisms. 733. The following statement may be considered as evident from the preceding: The volume of a triangular pyramid is definite, and is greater than that of any set of inscribed prisms, and less than that of any set of circumscribed prisms. 734. Theorem. The difference in volume between a set of inscribed prisms and the corresponding set of circumscribed prisms is the volume of the prism upon the base of the pyramid. 735. Theorem. The volume of a triangular pyramid is the common limit of the volumes of the set of inscribed prisms and the corresponding set of circumscribed prisms, as the number of these prisms is increased indefinitely. Given the triangular pyramid D-ABC, a set of inscribed prisms, and corresponding set of circumscribed prisms. To prove that volume of D-ABC is the common limit of these two corresponding sets of prisms. R N M C Outline of proof. The volume of the triangular pyramid is greater than the sum of the inscribed prisms and less than the sum of the circumscribed prisms. § 733 Therefore, the volume of the pyramid differs from either by an amount less than the volume of the prism upon the base of the pyramid. § Why? But the volume of this prism can be made as small as desired, that is, it has zero for a limit. A B § 485 (2) Hence, the volume of the set of inscribed prisms and the set of circumscribed prisms, each have the volume of the triangular pyramid for a limit. § 483 EXERCISES 1. Corresponding sets of inscribed and circumscribed prisms are formed in a triangular pyramid whose base has an area of 25 sq. in. Compute the series of differences between the corresponding sets when their altitudes are successively 0.1 in., 0.01 in., 0.001 in., and 0.0001 in. 2. The base of a pyramid is an equilateral triangle 3 in. on a side and its altitude is 7 in. Find the difference between the corresponding sets of inscribed and circumscribed prisms when the altitude of each prism is. 1000 of the altitude of the pyramid. 3. Could a quadrangular pyramid be used instead of a triangular pyramid in the discussion of §§ 730-733? 736. Theorem. The volumes of two triangular pyramids, having equivalent bases and equal altitudes, are equivalent. A A A' Given the triangular pyramids O-ABC and O'-A'B'C', having equivalent bases and equal altitudes; and, for convenience, having their bases in the same plane. To prove that VV', where V denotes the volume of O-ABC and V' the volume of O'-A'B'C'. Proof. Let the common altitude ON be divided into any number of equal parts, and pass planes through the points of division parallel to the plane of the bases. Construct the set of circumscribed prisms for each pyramid upon the sections formed by these planes. Then the sections of the pyramids made by each plane are equivalent. § 710 Let DEF and D'E'F' be the sections made by any one of these planes, and P and P' the corresponding prisms. Then volume of P= volume of P'. § 686 Let S denote the volume of the sum of the prisms in the set of which P is one; and S' that of which P' is one. And S and S' are variables that are equal for all their successive values as the number of the divisions of ON is increased indefinitely. 737. Theorem. The volume of a triangular pyramid is equal to one-third the product of its base and altitude. V=Bh. Given the triangular pyramid O-PQR. To prove that V=Bh, where V denotes the volume of the pyramid, B the area of its base, and h its altitude. Proof. Upon the base PQR construct a triangular prism NR of altitude h, and having its lateral edges parallel to OQ. The prism NR is divided into three triangular pyramids by the sections OPR and ONR. Pyramid R-NOM = pyramid O-PQR. § 736 Pyramid R-OPQ=pyramid R-ONP. § 736 But pyramid R-OPQ is the same as pyramid O-PQR. Therefore the three triangular pyramids are equal, and O-PQR is one-third the volume of prism NR. But volume of prism NR=Bh. .. V=}Bh. 738. Theorem. The volume of any pyramid is equal to one-third the product of its base and altitude. V=Bh. § 684 Why? Suggestion. From one vertex of the base draw all the diagonals of the base. Pass R planes through the vertex of the pyramid and each of these diagonals. All the pyramids have M the same altitude. Apply § 737 to each triangular pyramid formed and add the results. N 739. By a like treatment to that referred to in § 650, the truth of the following statement may be established: The volume of a circular cone is the common limit of the volumes of inscribed and circumscribed pyramids with regular polygons for bases, as the number of faces is indefinitely doubled. 740. Theorem. The volume of a circular cone is equal to one-third the product of its base and altitude. V=Bh=}πr2h. Given a circular cone. To prove V=Bh=r2h, where V denotes volume, B area of base, h altitude, and r radius. Proof. Inscribe a pyramid with a regular polygon for base, and let area of its base. M N h By doubling indefinitely the number of faces of the pyramid, V'→V, and B'→B. § 739, § 489 § 485 (2) But V'B'h, being variables that are always equal. § 738 Also .. V=Bh. В=πr2. .. V = {{πr2h. § 485 (1) § 498 § 111 Remark. It is to be noted that the formulas of this article apply to any circular cone. Compare with the remark of § 719. EXERCISES 1. Show that for any pyramid or circular cone B = = 2. Find the volume of a square pyramid whose base is 16 in. on a side, and whose altitude is 10 in. 3. Find the volume of a pyramid whose base is a regular hexagon 8 in. on a side, and whose altitude is 14 in. 4. Find the volume of a pyramid whose base is a square 3√2 in. on a side, and whose altitude is 63 in. |