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Cube

A rectangular solid is a three-dimensional figure formed by six rectangular surfaces, as shown below. 

Each square surface is a face.  Each solid or dotted line segment is an edge, and each point at which the edges meet is a vertex. A rectangular solid has six faces, twelve edges, and eight vertices. Edges means sides and vertices means corners. Opposite faces are parallel rectangles that have the same dimensions.

The surface area of a rectangular solid is equal to the sum of the areas of all the faces. The volume is equal to

        (length) × (width) × (height);

        in other words, (area of base) × (height).

  • In a cube, there are 6 faces, 8 vertices and 12 edges.
  • Vertices means corners and edges means sides.
  • Volume = a3.        Volume is always represented in cubic terms of the side
  • Surface Area = 6a2, where ‘a’ is the side of a cube. Surface area is always represented in square units.
  • Longest Diagonal = Length of the longest rod can be kept inside a cubical room = a3

                                                           

 

In the rectangular solid above, the dimensions are 3, 4, and 8. The surface area is equal to 2[(3 × 4) + (3 × 8) + (4 × 8)] = 136.  The volume is equal to 3 × 4 × 8 = 96.

 

Illustration 1 :

The side of a cube is 14 cms. Find its volume and surface area.

Solution

Volume = a3 = 14 × 14 × 14 = 2744 cubic cm
        Surface Area = 6a2 = 6 × 14 × 14 = 936 sq cm.

 

 

Illustration 2 :

The surface area of a cubical room is numerically equal to its volume. A rod has a length of 10 metres. Is it possible to keep that rod inside that room?

Solution

Surface Area = Volume ⇒ 6a2 = a3 ⇒ 6 = a. Side of the cubical room is 6 metres
Longest rod = a√3 ⇒ 6√3 ⇒ 6 × 1.732 ⇒ The product is more than 10.2 metres. It implies that a rod of 10 metres long can be kept inside that room.

 

Cuboid

In a cuboid, there are 6 faces, 8 vertices and 12 edges.

Volume = l × b × h, where l, b and h is length, breadth and height respectively.

Surface Area = 2(lb×bh × lh)

Longest diagonal = Length of the longest rod that can be kept inside that room is = l2 + b2 + h2

 

Illustration 3 :

A cuboid has length 10, breadth 15 and height 20m. Find its volume and surface area.
Solution: Volume = l × b × h = 10 × 15 × 20 = 3000 cubic units.
Surface Area = 2(lb + bh + lh) ⇒ 2(10 × 15 + 15 × 20 + 20 × 10) ⇒ 1300 square units.

Solution

Three cubes having an edge of 7 cm are placed together, thus resulting into a cuboid.  Find the volume and surface area of the cuboid.
Solution: When 3 cubes having a side of 7 cms each are placed together, then the length of the resulting cuboid is (7 + 7 + 7) = 21 cm, and the breadth and the length of the resulting cuboid will be same as each of the original cubes i.e. 7 each.
Volume = 21 × 7 × 7 = 1029 cubic cm.
Surface Area = 2 (21 × 7 + 7 × 7 + 7 × 21) = 686 sq. cm.





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