1. Two Dimensional Figures:
These are plane figures which just have measurements like length and breadth.
Example: Rectangle, Square, etc.
2. Three Dimensional Figures:
These are solid figures which have measurements like length, breadth, height.
Example: Cube, cuboid , etc.
Different views of three dimensional shapes:
Any 3-D object when viewed from different positions might appear different.
Example 1: Look at different views of glass from different positions.
Example 2: Look at different views of solid made of four cubes.
Example 3: Draw the top and side views for following objects:Solution:Â The top and side views for given figures are:
Example 4: Draw the front view, side view and top view for given figure:Â Solution:
Mapping Space around Us:
Map:
It is a figure which depicts the location of an object with respect to other objects.
In map, following are some important points that need to be noted:
(a) Symbols are used in maps to represent different objects, places, etc.
(b) Size of objects remain proportional to their actual size and not on the basis of perspective i.e. it does not make difference in size of object if its nearer of far from the observer.
(c) A scale is pre-fixed for every map. All the lengths are then drawn relative to that scale.
Example: Figure below shows a map for a particular college.
Faces, Edges& Vertices:(a) FaceÂ : These are the plane figures of which solids are made.
(b) EdgeÂ : These are the line segments where faces meet.
(c) Vertices/VertexÂ :Â These are the points where edges meet.
1. Polyhedrons:
All solids which have faces, edges and vertices are called polyhedrons.
2. Non â€“ Polyhedrons:
Those solids which either does not have faces, edges or vertices are called non - polyhedrons.
3. Convex Polyhedron:
A polyhedron in which the line segment formed on joining any vertices lies completely inside it is then such polyhedrons are called convex polyhedron.
4. Concave Polyhedron:
A polyhedron in which even if one or more line segment formed on joining any vertices lies outside it then such polyhedrons are called concave polyhedron.
5. Regular Polyhedron:
A polyhedron in which each face is a regular polygon and has same number of faces as vertices is called a regular polyhedron.
6. Irregular Polyhedron:
A polyhedron which is not a regular polyhedron called irregular polyhedron.
7. Prism:
It is a polyhedron in which base and top are the congruent polygons and whose sides are made of parallelograms.
8. Pyramid:
It is a polyhedron whose base is a polygon and the lateral faces are triangles with a common vertex.
9. Eulerâ€™s formula:
As per Euler, every polyhedron will satisfy the criterion F + V â€“ E = 2, where F is the number of faces of polyhedron, V is the vertices of the polyhedron and E is the number of edges of the polyhedron.
Example 1: Using Eulerâ€™s formula, find the number of faces if the number of vertices are 6 and the number of edges are 12.
Solution:Â Given, V = 6 and E =12.
We know Eulerâ€™s formula, F + V â€“ E = 2
So, F + 6 â€“ 12 = 2.
Hence, F = 8.
Example 2: Is it possible that a polyhedron have 10 faces, 20 edges and 15 vertices?
Solution:Â Given, F = 10, V = 15 and E = 20.
We know Eulerâ€™s formula, F + V â€“ E = 2.
If any object satisifies the formula, then it is a polyhedron. So, letâ€™s check it for given values.
LHS = 10 + 15 â€“ 20 = 5
RHS = 2.
Here, LHS â‰ RHS, thus it isÂ not possible to have such polyhedron.Â
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