(number of faces) + (number of vertices) - (number of edges) = 2 This formula Cylinder Right cylinder Oblique cylinder Truncated cylinder Cone Right cone
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[PDF] Math 366 Lecture Notes Section 114 – Geometry in - TAMU Math
The vertices of the polygonal regions are the vertices of the polyhedron, and the sides of each polygonal region are the edges of the polyhedron An oblique prism is one in which some of the lateral faces are not bounded by rectangles
[PDF] 111: Space Figures and Cross Sections Polyhedron: solid that is
The number of faces (F), vertices (V), and edges (E) of a polyhedron are related by the formula: Oblique Prism: lateral edges not perpendicular to the bases
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congruent, regular polygons that meet at all vertices in the same way prism - A polyhedron lateral edges of a prism – The intersection of two lateral faces of the prism oblique prism - A prism in which the lateral edges are not perpendicular
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In an oblique prism , the edges of the faces Faces = SA = Ph+2B prism Vertices = LA = Ph Edges = _ v= lwh Triangular prism Faces = _5 Faces = Vertices =
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*Slant height of a right cone The distance from the edge of the base It has 6 faces, 12 vertices, and 18 edges Cylinder A three-dimensionl figures with two
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numbers of faces, vertices, and edges of any polyhedron The result is known as surface area (of a prism) right prism oblique prism • cylinder bases, altitude
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the vertices and the sides of each Prism Def A prism is a polyhedron in which two congruent faces lie in parallel An oblique prism is a prism in which at
[PDF] LESSON 6 Geometry and Transformations Solid - Eduxuntagal
(number of faces) + (number of vertices) - (number of edges) = 2 This formula Cylinder Right cylinder Oblique cylinder Truncated cylinder Cone Right cone
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LESSON 6 Geometry and Transformations
Solid neometric figures
Polyhedra (A polyhedron is a solid with no curved surfaces or edges. Al1 faces are polygons and al1 edges are line
segments. The comer points where three or more adjacent sides meet are called vertices). Prisms:Cuboid Right pyramid
Pyramids:
Oblique pyramid Parallelepiped
Right pyramid Oblique pyramid Truncated pyramid (frustum of a pyramid)Regular polyhedra (Platonic solids)
i ,/ ' j l... , ' .g> -. V b --' Tetrahedron Hexahedron(Cube) Octahedron Dodecahedron IcosahedronEvery polyhedron has a dual polyhedron with faces and vertices interchanged. The dual of every Platonic solid
is another Platonic solid, so that we can arrange the five solids into dual pairs: The tetrahedron is self-dual (i.e. its dual is another tetrahedron). - The cube and the octahedron form a dual pair. - The dodecahedron and the icosahedron form a dual pair.One can construct the dual polyhedron by taking the vertices of the dual to be the centers of the faces of the
original figure. The edges of the dual are formed by connecting the centers of adjacent faces in the original. In
this way, the number of faces and vertices is interchanged, while the number of edges stays the same.
Euler's Formula (Polyhedra): (number of faces) + (number of vertices) - (number of edges) = 2 This formula is true for al1 - convex polyhedra - as well as many types of concave polyhedra.LESSON 6
Solids of revolution
Geometry and Transformations
(The solids of revolution are obtained by rotating-a plane figure in space about an axis coplanar to the figure).
Cylinder
Right cylinder Oblique cylinder Truncated cylinder Cone Right cone Oblique cone Truncated cone Frustum of a coneSphere fimmidai fmtum and ccmicñl fiustum.
Sphere
Open spherical sector
Spherical segment
Spherical cap
Spherical sector (spherical cone)
__-- - -- - .0 Spherical wedge
Ellipsoid
LESSON 6
Volumes
Geometry and Transforrnations
Cube: V = a3
Cuboid: V = length width height
Right prism (regular and irregular): V = b 1
Oblique prism (regular and irregular): V = b h
Right cylinder: v = b.l= n-r2 -1
Oblique cylinder: V = b h = n r2 h
b.hRight pyramid (regular and irregular): V = -
3 b.hOblique pyramid (regular and irregular): V = -
3 n.r2 .hRight cone: V =
3 n.r2 .hOblique cone: V =
3Frustum of a pyramid: V = hS(4 +$m +e)
3 n-h-(R2 + Rr+r2)Frustum of a cone: V =
34.n.r3
Sphere: V =
34-n-r, .r2 .r3
Ellipsoid: V =
LESSON 6 Geometry and Transformations
Sphere, cone and cylinder
Archimedes determined the ratio of the volume of a sphere to the volume of the circumscribed cylinder:
The volume of a cylinder is three times the volume of a cone with equal height and radius.The volume of a sphere is
two times the volume of a cone with equal height and radius.Prove:
if we consider solids with half-the volumes we can draw a semisphere, a cone with half the height and equal base, and a cylinder with half the height and equal base:
The horizontal section of half-the-cylinder is: In R' 1 a the volume of it is R d The horizontal section of half-the-cone is: as - = - the volume of it is