Cone - Ensui

Japanese: 円錐 - えんすい

If there is a circumference c on plane α and a point O outside α, then the surface formed by all the half-lines passing through any point on c with O as the edge is called a conical surface, O is the vertex, and each half-line is called a generating line. A solid body enclosed by a conical surface and plane α is called a cone (oblique cone), the circle on plane α is the base, and the length of the perpendicular line drawn from vertex O to plane α is called the height ( Figure (1)). The volume of a cone with a base radius r and height h is V = (1/3)π r ² h . If a cone is cut by a plane parallel to the base, the part of the cone between this plane and the base is called a truncated cone. When a straight line connecting the vertex O and the center of the base circle is perpendicular to the base, the cone is called a right circular cone ( Figure (2)). A right circular cone can also be considered as a solid body created by rotating an isosceles triangle OAB around the bisector of the vertex angle AOB. In this case, angle AOB is called the apex angle of the right circular cone, and the line segment formed by rotating OA is the generating line of the right circular cone. The lateral area of a right circular cone with a base radius of r and a generating line length of l is S = π rl .

[Minoru Kurita]

Cone (Diagram)
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Cone (Diagram)

Source: Shogakukan Encyclopedia Nipponica About Encyclopedia Nipponica Information | Legend

Japanese:

平面α上に円周c、α外に点Oがあるとき、Oを端とし、c上の任意の点を通る半直線全体のつくる面を円錐面、Oを頂点、各半直線を母線(ぼせん)という。円錐面と平面αとで囲まれた立体を円錐（斜円錐）といい、平面α上の円を底面、頂点Oから平面αへ下ろした垂線の長さを高さという（図の(1)）。底面の半径がr、高さがhの円錐の体積はV＝(1/3)πr²hである。円錐を底面に平行な平面で切り、この平面と底面の間にある円錐の部分を円錐台という。円錐で、頂点Oと底面の円の中心を結ぶ直線が底面に垂直のとき、この円錐を直円錐という（図の(2)）。直円錐は、二等辺三角形OABを、頂角AOBの二等分線を軸として1回転してできる立体とも考えられる。このとき、角AOBを直円錐の頂角といい、OAが回転してできる線分が直円錐の母線である。底面の半径がr、母線の長さがlの直円錐の側面積はS＝πrlである。