Plane and solid geometry . here,the sphere is said to be inscribed in the polyhedron. 928. Def. A polyhedron is inscribed in a sphere if all its vertices are on the surface of the sphere. 929. Def. If a polyhedron is inscribed in a sphere, thesphere is said to be circumscribed about the polyhedron. Ex. 1496. Find the edge of a cube inscribed in a sphere whose radiusis 10 inches. Ex. 1497. Find the volume of a cube: (a) circumscribed about asphere whose radius is 8 inches ; (6) inscribed in a sphere whose radiusis 8 inches. Ex. 1498. A right circular cylinder whose altitude is 8 inches is in-5c
Plane and solid geometry . here,the sphere is said to be inscribed in the polyhedron. 928. Def. A polyhedron is inscribed in a sphere if all its vertices are on the surface of the sphere. 929. Def. If a polyhedron is inscribed in a sphere, thesphere is said to be circumscribed about the polyhedron. Ex. 1496. Find the edge of a cube inscribed in a sphere whose radiusis 10 inches. Ex. 1497. Find the volume of a cube: (a) circumscribed about asphere whose radius is 8 inches ; (6) inscribed in a sphere whose radiusis 8 inches. Ex. 1498. A right circular cylinder whose altitude is 8 inches is in-5cribed in a sphere whose radius is 6 inches. Find the volume of thecylinder. Ex. 1499. A right circular cone, the radius of whose base is 8 inches^is inscribed in a sphere with radius 12 inches. Find the volume of thecone. Ex. 1500. Find the volume of a right circular cone circumscribedabout a regular tetrahedron whose edge is a. 426 SOLID GEOMETRY Proposition YII. Problem 930. To inscribe a sphere in a given tetrahedron. V. CMven tetrahedron V-ABC. To mscribe a sphere in tetrahedron V-ABC. I. Construction 1. Construct planes BABS, SBCT, and TCAB bisecting dihedral4 whose edges are AB, BC, and CA, respectively. § 691. 2. From 0, the point of intersection of the three planes,construct Oi^± plane ABC. § 637. 3. The sphere constructed with 0 as center and OF as radiuswill be inscribed in tetrahedron V-ABC. II. ProofAkgumext 1. Plane BABS, the bisector of dihedral Z AB, lies between planes ABV andABC] it intersects edge VC insome point as D, 2. .-. plane RABS intersects plane BCV in line BD and plane ACV in line AD, 3. Plane SBCT lies between planes BCV and ABC; i,e, it intersects plane RASBin a line through B between BA andBD, as Similarly plane TCAR intersects planeRABS in a line through A between Reasons1. Bv cons. 2. § 616. 3. By cons. 4. By steps sim-ilar to 1-3. BOOK IX 427 Argument AB and AD, as AR-, and plane SBCTintersects plane TCAR in a linethrough C as CT.
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Keywords: ., bookcentury1900, bookdecade1910, booksubjectgeometr, bookyear1912
