Class 6 – Mensuration

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Mensuration
  • Solids
  • Perimeter and area
  • Surface areas and volumes
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Introduction: In the world around us, we see objects having different shapes. Most objects that we can see or touch have one thing in common they all occupy space. In other words they all have three dimensions – length, breadth and height. Objects that we can feel and hold in our hands have three dimensions and so are referred to as 3D objects.

For an example the pen you hold, the book you are reading the chocolate you eat, etc. are called 3D objects.

3D means 3-dimensional shapes. Figures such as triangle, rectangle and square are called 2D or two dimensional shapes as they do not have depth (or) height they have only one surface. Let us recall some of the 2D and 3D shapes seen in the earlier classes let us revise key terms related to 3D shapes.

The flat surfaces of the solid figure are its faces. The top, bottom faces are called bases. The faces forming the sides are called lateral faces. The line segments where the adjacent faces of a solid meet are called edges. The point where the edges of a solid intersect is called a vertex.

Nets of solid shapes: A 2D figure that can be folded to give a 3D geometric solid is called the net of that solid.

The net is an arrangement of polygons with their edges joined. To manufacture these solids for packaging, we require the nets of these solids. To form the net of a cuboid, take a cuboidal carton (Take a box of cornflakes) and cut the box along the edge to lay it flat the 2D shape obtained is the net of the carton. It can be folded again to give the original 3D cuboidal box.

A dice is a very good example of a cube.

Net: A pattern that you can cut and fold to make a mode of a solid shape
Net of a cube:

    

The above figure shows one of the forms of the net of a cube which when folded forms the solid cube.

Shows below are the nets of a few more solids.

Net of a cuboid:

 

Net of acylinder:

             

Net of atriangular pyramid:

Net of atetrahedron:

Formula to find the number of faces, vertices and edges:  If n is the number of sides of the polygonal base then

1.   Pyramid is a solid, which forms a vertex on the top with a 2D shape as its base

2.   Prismis a solid which have both the faces (bottom, top) are same.

For a pyramid: number of the faces (F) = n + 1;

Number of vertices = n+ 1; number of edges = 2n

For a Prism: Number of faces = n + 2; number of vertices = 2n; number of edges = 3n

Faces of 3D shapes are polygons which are 2D shapes for example all the faces of a triangular pyramid are triangles and the faces of the cylinder include two circles.

Let us try and visualize.These 3D shapes on a 2D surface.

A tetrahedron is a 3D shape in which all four faces are triangular.

 

Euler’s Formula: For any polyhedron that does not intersect itself the number of faces plus the number of vertices minus the number of edges always equal to 2. This can be written as V + F – E = 2 (or) V+ F = E + 2 where F is number of faces, V is number of vertices and E is number of edges

Example: A cube has 6 faces 8 vertices and 12 edges

 6 + 8 – 12 = 2

Example: A tetrahedron has 4 faces, 4 vertices and 6 edges.

 F + V – E = 2  4 + 4 – 6 = 2

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Here are a few prisms and pyramids we use in daily life.