GEOMETRY. 



SURFACES OF SOLIDS. 

 The Parallelepiped Cube. 



The parallelepiped is a polyhedron, each of 

 whose faces is a 

 parallelogram ; if the 

 lateral edges are per- 

 each 

 right- 



pendicular to 



other, it is a 



angled parallelepiped. 



Then the surface of 



the figure is evidently found by taking the sum of 



the six rectangular faces. 



If the figure is a cube, we 

 have for its total surface six 

 times the surface of one face. 

 Let a = the edge of the cube, 

 then its surface = 6a*. 



The Prism. 



The surface of the 

 prism is evidently found 

 by taking the sum of the 

 surfaces of its different 

 faces. The lateral surface 

 is made up of rectangles 

 and the ends of triangles 

 or polygons. 



The Cylinder. 



If we consider the cylinder composed of thin 

 card-board, and that it is slit open along one of 

 its generating lines ; then if the surface be spread 

 out, it will be of the form of a rectangle, one of 



whose sides Aa is the circumference of the cylin- 

 der, and the adjacent side AB is the height of the 

 cylinder ; but the surface of this rectangle is 

 found by multiplying Aa by AB ; hence surface of 

 cylinder 



= Aa X AB = circumference X height. 

 This denotes the lateral surface only. We have 

 for the total surface to add to this the surface 

 of the two ends, which are circles, and conse- 

 quently can be found. 



The Pyramid. 



A pyramid is a polyhedron, bounded 

 polygon and triangular faces. 

 When the sides of the bounding 

 polygon are equal, the pyramid 

 is said to be regular. 



The total surface of the pyra- 

 mid will then be found by finding 

 the area of the polygon, and also 

 the areas of the triangular sides, 

 taking their sum. 



The Right Circular Cone. 



A conical surface is a curved surface, generated 

 by a moving straight line, which continually 

 touches a given curve, and passes through a given 

 fixed point not in the plane of the curve. The 

 solid bounded by a conical surface is a cone. If 

 the base is a circle, it is a circular cone ; and 

 when the vertex is in the straight line, passing 

 through the centre perpendicular to the plane of 

 the circle, it is called a right circular cone. 



If we conceive the cone made of thin card- 

 board, and slit open along one of its generating 



lines, it is evident that the figure thus formed is 

 the sector of a circle. 



Now area of sector ABC 



= \ arc BC X AB ; 

 .'. lateral surface of cone 



= \ circumference X slant height 



Hence the lateral surface of the cone is found 

 by multiplying half the circumference of the base 

 by the slant height. To find the total surface, we 

 have to add to this the area of the circular base. 



PROP. XV. The surface generated by a straight 

 line AB revolving about an axis X Y in its plane 

 is equal to the projection ab of 

 the line on the axis multiplied 

 by the circumference of the 

 circle, whose radius is the 

 perpendicular HO, erected at 

 the middle of the line, and 

 terminated by the axis. 



The surface generated by 

 AB round XY is evidently 

 that of the frustum of a cone 

 that is, it is the difference 

 of two cones, one of which 

 has Ed for the radius of its 

 base, and the other Aa. 

 Then this surface will evi- 

 dently 



= AB X cir., having HK as radius. 



But AB X cir., having HK as radius = a&X cir., 

 having HO as radius. Hence the proposition. 



The Sphere. 



Definition. A sphere is the sur- 

 face generated by the rotation of B 

 a semicircle, as ECF, about a 

 diameter, as EF. 



Let the arc AD be divided 

 into any number of equal parts. 

 The chords AB, BC, CD, &c., 

 when revolving round EF, gen- 

 erate surfaces which we can 

 determine ; for surface by 



637 



