Note on Solid Shapes and Their Nets

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Solid-Shapes

Solid shapes are there dimensional. Solids can be described in various types. Some have a flat surface, some have curved surface and some have both flat and curved surface.

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Nets for Solid figure

A pattern that you can cut and fold to make a model of a solid shape are a net of a solid figure.

Cube and Cube net

A cube is an object which looks like solid box-shaped that has six identical square faces.

c

A cube is a special case of a cuboid in which all six faces are squares i. e. lenght = breadth = height

Let each side of cube be a units

Then, total surface area of a cube =2(lb+lh+bh)

= 2(a x a + a x a + a x a)

= 2(a2 +a2 +a2)

= 2 x 3a2

= 6a2

= 6(side)2

Similarly, lateral surface area of a cube = 2h(l +b)

= 2a(a +a)

= 2a x 2a

= 4a2

= 4(side)2

and volume of a cube = l x b x h

= a x a x a

= a3

= (side)3

Cuboid and Cuboid net

Cuboid is a solid figure bounded by six faces. A cuboid has 6 rectangular faces. The opposite rectangular plane surfaces are identical. So, it is also called rectangular prism.

 Cuboid net.

The surface of a cuboid consists of six faces which are rectangular in shape. We can categories them in pairs of opposite faces as follows:

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  1. Front face ABGH and back face EFCD
  2. Top face EFGH and bottom face ABCD
  3. Side faces ADEH and BCFG
    These opposite faces are congruent.

Let l, b and h represent the length, breadth and height of the cuboid respectively. To calculate the surface area of the cuboid we need to first calculate the area of each face and then add up all the areas to get the total surface area.

Total area of top and bottom surfaces is lb + lb =2lb

Total area in front and back surfaces is lh + lh = 2lh

Total area of two side surface is bh + bh =2bh

Thus, total surface area of cuboid = 2lb+2lh+2bh

=2(lb + lh +bh)

The lateral surface area of a cuboid includes a front face, back face, and two side faces.

Therefore, lateral surface is of a cuboid = 2lh + 2bh

= 2h(l+b)

Also, volume of a cuboid = l x b x h

Cylinder and Net of Cylinder

A cylinder has two circular plane surfaces, one at its base and another at its top. It has a curved surface in the middle.

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To find the surface area of the cylinder, add the surface area of each end plus the surface area of the side. Each end is the circle so the surface area of each end isπr2, where r is the radius of the end.

There are two ends so their combined surface area is 2πr2. The surface area of the side is the circumference times the height or 2πr x h where r is the radius of the end and h is the height of the cylinder.

Therefore, curved surface area = 2πrh

and total surface area = 2πr2 = 2πrh = 2πr(r+h)

The base of a cylinder is a circle, with area πr2

The volume of a cylinder is, therefore,

πr2 x height = πr2

Cone and Net of Cone

A cone is a closed figure with a plane surface and a curved surface.

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Pyramid and Its Net

There are three different solids that you can make with triangles. Mathematicians use the word tetrahedron to describe a triangular pyramid. Since it uses squares as well as triangles.

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Tetrahedron

Tetrahedron is a triangular pyramid having congruent equilateral triangular for each of its faces.

The total surface area of a tetrahedron

= 4 x area of an equilateral triangle

= 4 x \(\frac{√3}{4}\) side2

=√3 side2

Volume of a tetrahedron = \(\frac{1}{3}\) area of base x height of pyramid

Net of tetrahedron

 Net of tetrahedron

Square pyramid

A square pyramid is a pyramid having a square base.

In the figure, ABCD is a square base PO is the height of the pyramid. M is the mid-point of BC.

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PM is the slant height of the pyramid.

Lateral surface area of a square pyramid = area of triangular faces

= 4 x area of Δ PBC

= 4 X \(\frac{1}{2}\) x BC X PM

= 2 X BC X PM

Total surface area of the square pyramid = LSA + area of base.

Also, volume of the square pyramid = \(\frac{1}{3}\) area of base x height

= \(\frac{1}{3}\) A x h

Net of square pyramid

Net of square pyramid

 

  • Solid figures are three- dimensional objects.
  • A cube is a 3-dimensional solid figure that bounded by the square faces.
  • A solid object with two identical flat ends that are circular and one curved surface is a Cylinder.
  • The base of a pyramid may be any polygon. If the base is a triangle, the pyramid is called tetrahendron.
  • A cone is a solid object with the flat surface which is circular and one curved surface.
  • A triangular prism is a prism whose bases are triangles and whose lateral faces are rectangles.
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  •  A rectangular block 8cm high has a square base of area 100cm2. Find its total surface area.

     500cm2
     450cm2
     520cm2
     550cm2
  • The surface area of a cube is 726cm2, find the volume of the cube.

     1331cm3
     1221cm3
     1200cm3
     1232cm3
  • Find the volume and total surface area of a cuboid whose length is 12cm, breadth is 10cm and height is 8cm.

     950cm3, 550cm2
     960cm3, 592cm2
     955cm3, 555cm2
     900cm3, 590cm2
  • The height of a room is 3.5m and its contains 70m3 of air. Find the area of carpet required to cover its floor.

     25m2
     33m2
     20m2
     22m2
  • A rectangular block has a square base of area 900cm2. If its total surface area is 3600cm2, find its height.

     17cm
     22cm
     12cm
     15cm
  •  A room is 10m long, 6m wide and 5m high. Find the cost of painting its four walls at Rs 5 per square meter.

     Rs 800
     Rs 980
     Rs 550
     Rs 750
  • The total Surface area of a cube is 216cm2. Find its volume.

     222cm3
     210cm3
     275cm3
     216cm3
  • Find the length of each edge of a cube, if its volume is: 125cm3.

     12cm
     10cm
     5cm
     7cm
  • A room is 12m long, 8m wide and 5m high. What will be the cost of painting its four walls at Rs 10 per m2.

     Rs 2000
     Rs 1500
     Rs 2500
     Rs 1000
  •  Find the volume and total surface area of a cube whose each edge is, 6cm.

     216cm3, 216cm2
     515cm3, 515cm2
     250cm3, 250cm2
     150cm3, 150cm2
  • The total surface area of a cube is 150 m2. Find its volume.

     129m3
     175m3
     150m3
     125m3
  • The volume of cuboid is 3456cm3. If the length and breadth of the cuboid are 24cm and 18cm respectively, find its breadth

     12cm
     6cm
     15cm
     8cm
  •  Find the surface area of a box opened from the top whose shape is like a cube of edge 20cm.

     3000cm2
     2000cm2
     2500cm2
     1500cm2
  • Find the length of each edge of a cube, if its volume is, 512cm3.

     6cm
     10cm
     12cm
     8cm
  • How many bricks are required to build a wall 20m long, 2.5m high and 25cm thick if one brick measures 20cm long, 10cm wide and 5cm thick?

     12500
     12550
     12775
     12750
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Anmol

Net of rectangle based pyramid


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Horizonkc

The breaks is 5cm by 3cm by and 2cm and the box is 22cm by 16 cm and 14cm. How many bricks can be kept in the box. Find in six ways.


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Dima

Find thearea of shaded part of circle


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