In this page provided formulas for Surface Area, volume, Diagonal Length, Perimeter of Cuboid and Cube with examples

## Surface Area and volume of cuboid and Cube Formulas with Examples

### Cuboid

A solid which contains length, breadth and height is called *cuboid.* A cuboid is three dimensional box. It is virtue of its length (l), breadth ( b) and height (h).

### Cube

A *cube* is a cuboid which has all its edges equal

i,e length = breadth = height = a

A cuboid or cube have 6 faces, 8 vertices and 12 edges.

#### Perimeter of Cuboid and Cube

The perimeter of the cube or cuboid gives the complete edge to edge separation of the shape of the same

*Perimeter of Cuboid *= **4 ( l + b + h)**

*Perimeter of cube* = **4 ( a + a + a ) = 12a**

#### Surface Area of Cuboid and Cube

The area of four faces of cube or cuboid is called the lateral surface area of the same.

Lateral Surface area of cuboid = lh + lh + bh + bh = 2 (lh + bh) = 2 (l +b)h

Lateral Surface area of cube = aa + aa + aa + aa = 4a^{2}

Six rectangular pieces are covered by cuboid total surface area.

Total surface area of cuboid = Total area of the all rectangular pieces = lb + lb + bh + bh + lh + lh

**Area of Cuboid = 2 ( lb + bh + lh)**

Total surface area of cube = Total area of the all square pieces = aa + aa + aa + aa+ aa + aa

**Area of Cube = 6a ^{2}**

#### Volume of Cuboid and Cube

Cuboid or cube measure of this occupied space is called the volume of Cuboid and Cube.

The total surface area of cuboid = The area of plan rectangle occupied by each rectangle x height = Length x Breadth x Height

** Volume of the Cuboid =( l x b x h) **cubic units

**Volume of Cube = edge x edge x edge = a x a x a = a ^{3 }**cubic units

#### Diagonal length of Cuboid and Cube

Diagonal a Cuboid (d) =

Diagonal a Cube (d) =

__Formulas of Cuboid and Cube__

Description |
Cuboid |
Cube |

Perimeter | 4 ( l + b + h) | 12a |

Lateral Surface area | 2 (l +b)h | 4a^{2} |

Total surface area | 2 ( lb + bh + lh) | 6a^{2} |

Volume | ( l x b x h) | a^{3} |

Diagonal Length |

#### Examples of Cuboid and Cube

**Ex – 1 :** The required rectangular box has length, breadth and height as 80 cm, 40 cm and 20 cm respectively. Then how many Square sheets are required to prepare rectangular box with side of 40 cm.

**Solution:** Here rectangular box

Length = l = 80 cm ,

Breadth = b = 40 cm &

Height = h = 20 cm

Side of the square sheet = a = 40 cm

Total surface area of the rectangular box = 2 (lb + bh + lh ) = 2 [ (80 x 40) + (40 x 20) + (80 x 20)] = 11200 cm^{2}

The surface area one single square sheet = 2 a^{2} = 2 x 40 x 40 =1600 cm^{2}

Therefore, number of square sheets required for prepare a rectangular box = Total surface area of the rectangular box / Total surface area of the rectangular box

= 11200 / 1600 = 7

Hence 7 sheets are required.

**Ex-2 :** The length of a rice godown is double its breadth. Its height is 3 m. The area of four walls is 108 m^{2}. Find the volume of the godown.

**Solution:** Take godown breadth = b m then length = l = 2b m, height = 3 m

Area of four sides = 2(l+b)h = 108

⇒ 2 ( 2b + b) 3 = 108

⇒ b = 6 m

⇒ l = 2b = 12 m

Volume of rice godown = 12 x 6 x 3 = 216 m^{3}

**Ex-3** : A solid cube of lead having edge measures 44 cm. How many spherical balls can be made by using solid cube of lead and each ball being 88 cm in diameter.

**Solution:**

Side of the solid cube lead = a = 44 cm

Volume of the solid cube = a^{3} = 44^{3}

Radius of spherical balls = r = 88/2 = 44 cm

Volume of each spherical ball = = ( 4/3) x (22/7) x (3 x 3 x 3 )

Number of spherical balls = Volume of the solid cube/Volume of each spherical ball

= 44 x 44 x 44 x 3 x 7 / ( 4 x 22 x 44)

= 462 nos.

**Ex-4 :** Two cubes volume 64 cm^{3} each are joined end to end. Find total surface area of the resulting cuboid.

**Solution:** Take side of the cube = a

Then Volume of cube = a^{3 }= 64 cm^{3 }

⇒ a = 4 cm

Now resulting cuboid length = 4 + 4 = 8 cm , breadth = 4 cm & height = 4 cm

Surface Area of the cuboid = 2 ( 8×4 + 4×4 + 4×8) = 160 cm^{2}

**Ex-5 **: The ratio of curved surface area and total surface area of a cube with edge ‘a’ cm is

**Solution:** Curved surface area of cube = 4a^{2}

Total surface area of cube = 6a^{2}

So ratio of curved surface area to total surface area of cube = 4 : 6 = 2 : 3

**Ex-6 :** Find the diagonal of a cuboid with length = 2a cm, breadth = 3a cm and height = cm

**Solution:** Cuboid have length (l), breadth ( b) and height (h).

then Diagonal a Cuboid (d) =

=

= 5a cm

**Ex-7 :** A cubical solid block of metal 49 cm x 44 cm x 18 cm is melted and formed into a solid sphere . Then find the radius of the sphere.

**Solution**: Here Volume of solid block = Volume of solid sphere

Let radius of sphere = r cm

⇒ 49 x 44 x 18 = (4/3) x (22/7) x (r^{3})

⇒ r = 21

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