The same questions arise concerning faces and vertices, and it's even harder to decide in those cases. Does a cone have a vertex? A year later, we got this question along similar lines: Does a Cone have an Edge? A Vertex? Our 4th grade math textbook defines a cone as "A solid figure with one circular face and one vertex. The textbook defines a face as "A flat surface of a solid. If a cone has only one face, then it can't possibly have an edge. Therefore, if it doesn't have an edge, it can't have a vertex.
I responded again, after referring to that previous answer: Elementary texts and high school texts, for that matter are not always very careful about definitions. The problem really is that the same word can be used with slightly different but related definitions, and we don't always bother to specify how to modify the definitions when we move to a different context.
The definitions given are for a polyhedron. When you talk about a cone or cylinder, you have to either use a different set of words , since "edge" and "vertex" as defined don't apply at all, and "face" applies only to one of the two surfaces of a cone; or you have to modify the definitions to allow curved edges and faces.
Taking the latter approach, the cone will have two faces, one curved, and one curved edge. I'm not sure I've ever seen such modified definitions actually stated, but I have no trouble allowing them, as long as we state them clearly!
When they then go ahead and say it does have an edge or a vertex, children are bound to be confused. The really tricky part here is that the "vertex" of a cone has nothing to do with edges, so it needs a whole new definition ; and I can't think of a really good elementary-level definition for what they obviously mean, which is simply a "point.
Does a cylinder have edges? A month later we got yet another question about this, which got a long answer: Number of Cylinder Edges My 8-year-old son was asked "how many edges are there on a solid cylinder?
His answer was "2" and it was marked as incorrect. He truly believes in his answer and has asked for my assistance in researching. I answered first, again having referred to the previous answer as background: It depends on how "edge" was defined in his class, which may not agree with his intuitive definition. Often, an edge is required to be straight , in which case a cylinder has no edges. Unfortunately, elementary texts are not always very careful about definitions, and they can ask questions like this that are really worthless.
The only definition of "edge" that would make sense in this context would be the one your son is naturally using a boundary between smooth surfaces making up an object , which would allow a cylinder to have two edges.
Asking this question with the other definition only invites confusion, so I wish they wouldn't ask it. I'd like to hear how they did define the word. What is the purpose of definitions? But this discussion went further, outside of Ask Dr. I will just quote pieces of it that relate most directly to our present issue of definition: This is a common issue among elementary teachers, and some elementary text book writers.
Basically different sources put down different answers. The underlying issue is: What is the context? What is the larger mathematics one wants to engage with? Without this, there are too many plausible responses. However, some elementary texts and test writers decide they know best and give distinct definitions of 'faces', 'edges', and 'vertices'. When doing so, there should be some good mathematical reason for doing that. Some set of situations one is trying to make sense of.
Simple extrapolation on one basis or another, without investigating the good and bad patterns, is a source of trouble. That, unfortunately, routinely happens in elementary and some high school materials. If faces are 'flat regions' and 'edges' are straight lines , then a cylinder has two faces, no edges, and there is no real purpose in the answer.
The eccentricity of a circle is zero, because the focal points are in exactly the same place the centre we also say that they are coincident.
The distance from the centre to the focal point is therefore zero. The eccentricity increases as the ellipse becomes longer, but is always less than 1. When the distance from the centre to the focal point is the same as the distance from the centre to the vertex, then the ellipse has become a straight line and its eccentricity is equal to 1. Parabolas and Hyperbolas are more forms of curved shapes, but they are more complicated to define than circles and ellipses.
They are closely related to each other and to circles and ellipses, because they are all conic sections , i. The characteristics of conic sections have been studied for millennia and were a subject of interest for ancient Greek mathematicians such as Euclid and Archimedes.
The diagram below shows a double cone, rather like a sand-timer. If the plane cuts the cone at an angle parallel to the base of the cone i. If the plane cuts the cone parallel to the side of the cone , then a parabola is formed centre. If the plane cuts the cone at an angle between these two, such that it maintains contact with the sides of the cone in all locations, then an ellipse is formed bottom left. If the plane cuts through both cones at a more vertical angle, then the section is a hyperbola.
All parabolas have the same characteristic shape, no matter how big they are. As you zoom out further and further from the vertex towards infinity, the parabola changes from a bowl shape to a hairpin shape, with its arms becoming closer and closer to parallel. Unlike parabolas, hyperbolas can be different shapes , because the angle of the cut can vary widely.
Circles are part of basic geometry, and you really need to know how to calculate basic properties of them. It is, however, probably unlikely that you would need to do more than be aware of the existence of the other shapes unless you wished to get seriously into engineering, physics, or astronomy.
That said, you may find that you appreciate knowing that the concave curves of a power station cooling tower, or the light from a downward-pointing halogen lamp, are in the shape of a hyperbola.
This eBook covers the basics of geometry and looks at the properties of shapes, lines and solids. These concepts are built up through the book, with worked examples and opportunities for you to practise your new skills.
All of these figures are curved in some way, so they have no edges or vertices. What about their faces? A sphere has no faces, a cone has one circular face, and a cylinder has two circular faces.
Therefore, the number of faces increases by one from one figure to the next. A cone has one face. It is a three-dimensional shape with a circular base, one side and one vertex. Faces can be identified as the flat surfaces on a three-dimensional figure. Definition of a Polygon. These figures have curved surfaces, not flat faces. A cylinder is similar to a prism, but its two bases are circles, not polygons.
Also, the sides of a cylinder are curved, not flat. A cone has one circular base and a vertex that is not on the base. They will talk about how many sides a 2D shape has , and whether the sides are straight or curved. It is very important that they handle 3D shapes in order to be able to count their faces , edges and vertices, so they will probably construct their own 3D shapes from nets.
A square and a rectangle each have 4 sides and 4 angles. The only difference between them is that a square is a rectangle with equal sides. Both have 4 90 degree right angles. A triangle has 3 sides and 3 angles. Its proper name is a "tetrahedron".
The tetrahedron has the extra interesting property of having all four triangular sides congruent. An Egyptian pyramid has a square base and four triangular sides. A cyclic pentagon is one for which a circle called the circumcircle goes through all five vertices.
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