Plane and solid geometry . , i,e. if the base of thespherical sector is a zone of one base, the spherical sector issometimes called a spherical cone. Thus if circular sector AOB (Fig. 1) revolves about diameterCD as an axis, arc AB will generate a zone which will be thebase of the spherical sector generated by circular sector AOB(Fig. 2). If circular sector 7>*OC revolves about diameter CD^the spherical sector generated, whose base is the zone generatedby arc BC^ will be a spherical cone (Fig. 3). BOOK IX 463 1004. Cor. ni. The volume of a spherical sector isequal to the product of its base
Plane and solid geometry . , i,e. if the base of thespherical sector is a zone of one base, the spherical sector issometimes called a spherical cone. Thus if circular sector AOB (Fig. 1) revolves about diameterCD as an axis, arc AB will generate a zone which will be thebase of the spherical sector generated by circular sector AOB(Fig. 2). If circular sector 7>*OC revolves about diameter CD^the spherical sector generated, whose base is the zone generatedby arc BC^ will be a spherical cone (Fig. 3). BOOK IX 463 1004. Cor. ni. The volume of a spherical sector isequal to the product of its base and one third the radiusof the sphere. Outline of Proof Let V denote the volume generated by polygon OA^B^c\ v thevolume generated by polygon OABC, s the area of the surfacegenerated by broken line abc, s thearea of the surface generated bybroken line ABCj and Z the base ofthe spherical sector generated by cir-cular sector AOC. Then V = S -^R, and v = s^^a. V 8 a a s i?2 But ~ = ^^ (Args. 2-5, § 969).s a- ^\R ,2 — «.3. • . V a- a a Then by steps similar to § 996, b and c, and § 997, thevolume of the sphericqj sector generated by circular sectorAOC= Z -^R. 1005. Cor. IV. If V denotes the volume of a sphericalsector, z the area of tlw zone forming its base, H the alti-tude of the zone, and R the radius of the sphere, V=ZiR (§1004) = (^H2irR)\R (§979) Ex. 1600. Considering the earth as a sphere with radius i?, find thevolume of the spherical sector whose base is a zone adjoining the north Is the one volume twice the other ? 7? 2 7? pole and whose altitude is —; —^ 3 3 Compare your results with those of Ex. 1558. Ex. 1601. Considering the earth as a sphere with radius i?, find thevolume of the spherical sector w^hose base is a zone extending: (a) 30°from the north pole; {h) 60° from the north pole. Is the one volumetwice the other ? Compare your results with those of Ex. 1559. 464 SOLID GEOMETRY Ex. 1602. Considering the earth as a sphere \vith radius i?, find thev
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