Class 8 Maths Mensuration | Solid Shapes |

__Solid Shapes__

Many objects that we see in our day to day life like book, pencil box, ice cream cone, football and cylinder are three-dimensional objects (solid shapes). All these objects occupy some shape and have three dimensions- length, breadth and height or depth.

Some shapes have two or more than two identical (congruent) faces. For example the cylinder has congruent circular faces that are parallel to each other.

__Surface area of Cube, Cuboids and Cylinder__

The surface area of a solid is the sum of the areas of its faces.

**Cube: **

If each side of the cube is l, then total surface area = 6l^{2}

Example: Let each side of the cube is 5 cm,

So, total surface area = 6 * 5^{2 }= 6 * 25 = 150 cm^{2}

**Cuboid:**

Let h, l and b are the height, length and width of the cuboid respectively.

So, total surface area = 2 (h * l + b * h + b * l) = 2(lb + bh + hl)

Example: Let the height, length and width of the box shown above are 10 cm,

15 cm and 20 cm respectively.

Then the total surface area = 2 (10 * 15 + 15 * 20 + 10 * 20)

= 2 (150 + 300 + 200)

= 2 * 650

= 1300 m^{2}

**Cylinder:**

Let r is the radius of circular region and h is the height of cylinder.

Then total surface area of cylinder = 2πr(h + r)

__Problem:__ A closed cylindrical tank of radius 7 m and height 3 m is made from a sheet of metal. How much sheet of metal is required?

Solution:

Given: Radius of cylindrical tank (r) = 7 m

Height of cylindrical tank (h) = 3 m

Total surface area of cylindrical tank = 2πr(h + r)

= 2 * 22/7 * 7(3 + 7)

= 44 * 10

= 440 m^{2}

Hence, 440 m^{2} metal sheet is required.

__Volume of Cube, Cuboids and Cylinder__

Amount of space occupied by a three dimensional object is called its volume.

**Cube:**

The cube is a special case of a cuboid, where l = b = h.

Hence, volume of cube = l * l * l = l^{3}

__Problem:__ Find the volume of cube if its each side is of length is 5 cm.

Solution:

Given, l = b = h = 5 cm

volume of cube = l * b * h = 5 * 5 * 5 = 125 cm^{3}

** **

**Cuboid:**

Volume of a cuboid is equal to the product of length, breadth and height of the cuboid.

Let l is the length, b is the breadth and h is the height of the cuboid.

Then volume of cuboid = l * b * h

__Problem:__ Find the height of a cuboid whose base area is 180 cm^{2} and volume is 900 cm^{3}?

Solution:

Given: Base area of cuboid = 180 cm^{2} and Volume of cuboid = 900 cm^{3}

We know that, Volume of cuboid = l * b * h

=> 900 = 180 * h [Base area = l * b = 180 (given)]

=> h = 900/180

=> h = 5 m

Hence, the height of cuboid is 5 m.

**Cylinder:**

Let r is the radius of circular region and h is the height of cylinder.

Then volume of cylinder = πr^{2} * h = πr^{2}h

__Problem: __Find the height of the cylinder whose volume if 1.54 m^{3} and diameter of the base is 140 cm.

Solution:

Given: Volume of cylinder = 1.54 m^{3} and Diameter of cylinder = 140 cm

Radius (r) = 140/2 = 70 cm

Volume of cylinder = πr^{2} h

=> 1.54 = 22/7 * 0.7 * 0.7 * h

=> 1.54 = 22 * 0.1 * 0.7 * h

=> 1.54 = 1.54 * h

=> h = 1.54/1.54

=> h = 1 m

Hence, the height of the cylinder is 1 m.

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