1. Visualizing Solid Shapes

    The figures such as triangles, rectangle, squares, circles etc. which have only two dimensions, namely length and breadth are called plane figures or 2-dimensional figures.

Plane figures (2-dimensional figure)

The figure such as cube, cuboid, cylinder, pyramid etc. which have three dimensions namely length, breadth & height are called solid figures or 3-dimensional figures.

Solid figures (3-dimensional figure) 

Types of Solid  
    There are mainly three types of solids :
(1)     Prism :  A solid whose top and base are identical polygons and side faces are rectangles, is called a prism. In a square prism whose top and base are congruent squares. 

(2)     Pyramid : A solid whose side faces are triangles and base is any polygon is called a pyramid. Figure shows a pentagonal pyramid.

(3)     Sphere : Sphere is a solid whose every point is equidistant from a fixed point. Figure shows the sphere.


Note : (i)     All the side faces of a pyramid (squares, rectangular, triangular, pentagonal etc.) are triangular.
           (ii)     A  pyramid is named according to the shape of its non-triangular face. If all the faces are triangular, then it is called a triangular pyramid or tetrahedron.


1. Visualizing Solid Shapes

Chapter -15

Visualizing Solid Shapes

Measurement of the shapes from a certain angle or direction is called dimension.
The closed figures drawn on a flat-surface or on a plane are two-dimensional shapes. The sides of these shapes can contain straight or curved surfaces. The shapes can have 'n' number of sides.
2-D shapes are also called as plane figures
Three-dimensional shapes
Our entire world is made up of objects that occupy space. Any object or a shape that occupies space is three-dimensional.
The 3-D figures are also called as solid figures.
The flat surface of a solid shape is its face. A solid shape can have 'n' number of faces.
The line segment on a solid shape is called an edge.
The point where two or more edges join is a corner or vertex of a solid shape.
Oblique sketching is a pictorial representation of an object, in which the diagram is intended to depict the perspective of objects in three dimensions.
Isometric sketch is similar to oblique sketching but the solid figure is represented on an isometric sheet.
An isometric sheet will have dots in small equilateral triangles all over the sheet. The distance between each dot in the graph is usually considered as scale of 1cm.
A cross-section is the new face of the object we get when we slice the object from any direction. The newly formed face will always resemble a two-dimensional shape.
Shadow play is one of the ways of viewing solids. When light falls on an object, the resultant shadow thus formed is two-dimensional.

2. Faces, Edges and vertices

Number of faces, edges and vertices of some solids
1.    Cuboid    

A closed solid shape that has six rectangular surface is called a cuboid.
    (i)     The 8 corners of the cuboid are its vertices.
    (ii)     The 12 line segments that form the skeleton of the cuboid, are its edges.
    (iii)     The six flat rectangular surface that are the skin of the cuboid are its faces.

2.    Cube    
    A cube is a special kind of cuboid whose length, breadth, and height are equal.
    (i)     The 8 corners of the cube are its vertices.
    (ii)     The 12 line segments that form the skeleton of the cube are its edge.
    (iii)     The six flat rectangular surface that are the skin of the cube are its faces.


3.    Cylinder                                       
    A cylinder is a solid shape whose top and bottom are circular while the rest of the surface is curved.
    (i)     Number of faces is 3.
    (ii)     Has no vertex.


4.    Cone
    A cone is a solid shape having a plane circular end as the base and whose lateral surface is the curved surface tapering into a point, is called the vertex of the cone.
    (i)     Number of faces is 2.
    (ii)     No. of vertex 1
    (iii)     No. of edges 1

5.    Sphere
    A solid (3D) shape that has only a curved surface is called sphere.
    (i)     Number of faces is 1.

6.    Prism
    A prism is a solid whose base are identical polygonal shapes (Triangles, Quadrilateral, Pentagons) and other faces are rectangles.

7.    Pyramid 
    A pyramid is a solid whose base is a flat polygonal surface and whose side faces are triangles having a common vertex outside the surface of the base.


3. Nets for building 3-D shapes

NETS for Building 3D-shapes
    If we join together six identical squares, edge to edge, we get a cube or the outside surface of the cube.

 A net for a three dimensional shape is nothing but a sort of skeleton-outline in 2-dimension which, when folded, results in three dimensional shape. 

In figure (i) the shape of the pattern of six squares. When this shape is folded along the edges, a cube is formed as shown.


4. Drawing solids on a flat surface

Eulers Formula
    The number of faces, edges and vertices of prism and pyramids are connected by the formula :


where V, F and E stand for the number of  vertex , face and edge. This formula is known as Euler’s formula.    
    Cuboid :                 
    A  cuboid has                                      
    (i)     6 rectangular faces                
    (ii)     12 edges            
    (iii)     8 vertices.                               
      Let  F, E and V denote respectively the number of  faces, edges and vertices of a cuboid.
    Then, F – E + V = 6 – 12 + 8 = 2.  

Cube (Square prism) : 
     A  cube has                                                                    
    (i)     6 square faces    
    (ii)     12 edges
    (iii)     8 vertices
    Here, F = 6, E = 12 and V = 8
    F – E + V = 6 – 12 + 8 = 2.

Triangular Pyramid : A triangular pyramid (Tetrahedron) shown in fig. has :     (i)     4 faces 
    (ii)     6 edges
    (iii)     4 vertices
        Here, F = 4, E = 6 and V = 4
           F – E + V = 4 – 6 + 4 = 2

Illustration 1
        A polyhedron has 20 edges and 10 vertices. How many faces does the polyhedron have ?
        Here, E = 20, V = 10
        V + F – E = 2
       10 + F – 20 = 2
        F = 2 – 10 + 20 = 12
    Hence, the polyhedron has 12 faces. 

Illustration 2
        Using Euler’s formula find the unknown.

        We know that using Euler’s formula
            F + V – E = 2
        (a)     F + 6 – 12 = 2 
        or     F – 6 = 2 or F = 8
            Thus number of faces = 8
        (b)     5 + V – 9 = 2
            Thus number of vertices = 6
        (c)     2 + 12 – E = 2 
        or     32 – 2 = E or 30 = E
        Thus number of edges = 30

Illustration 3    
        Can a polyhedron have 10 faces, 20 edges and 15 vertices ?
        The given data will be of a polyhedron, if it satisfies the Euler’s formula.
        i.e.,    V + F – E = 2
        Here V = 15, F = 10 and E = 20
        Putting in L.H.S.
            15 + 10 – 20 = 5 RHS
            LHS RHS
        Hence given vertices, faces and edges can’t be that of a polyhedron.


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