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The prisms are formed with two parallel congruent polygonal bases. The lateral faces of a prism are parallelograms. In the case of a right prism, the lateral surfaces are all rectangles.

A rectangular prism or a cuboid has rectangles as bases. It has six rectangular faces, consisting of two base rectangles and four lateral rectangles. It has 12 edges and 8 vertices.

A cube is a special type of rectangular prism with all six square faces. Just as a rectangular prism, a cube also has 12 edges and 8 vertices.

**Net for a cube: **If a cubic box is cut suitably at the edges and laid flat, a pattern for a cube can be seen. This is called the net for the cube. The net pattern can be folded along the edges as shown and made into a cube without any overlap.

**Surface area of a cube using net**:

The surface area of a prism = 2 base area + lateral surface area.

The net of the cube shows that the surface of the cube contains six congruent squares. Any of the two opposite squares can be termed as the bases, while the remaining vertical square faces form the lateral surface. Hence the surface area of a cube is six times the area of its base. If the length of an edge is a units then the area of the base square = $a^{2}$ sq units.

**The surface area of a cube - Formula**

Surface area of cube of edge length a units = 6$a^{2}$ sq.units.

Areas are expressed as square units and volume as cubic units.Inversely if the surface area 'A' of a cube is known, the length of the edge is given by,

Length of the edge of a cube =$\sqrt{\frac{s}{6}}$ units.

A rectangular prism or a cuboid has rectangles as bases. It has six rectangular faces, consisting of two base rectangles and four lateral rectangles. It has 12 edges and 8 vertices.

A cube is a special type of rectangular prism with all six square faces. Just as a rectangular prism, a cube also has 12 edges and 8 vertices.

The surface area of a prism = 2 base area + lateral surface area.

The net of the cube shows that the surface of the cube contains six congruent squares. Any of the two opposite squares can be termed as the bases, while the remaining vertical square faces form the lateral surface. Hence the surface area of a cube is six times the area of its base. If the length of an edge is a units then the area of the base square = $a^{2}$ sq units.

Surface area of cube of edge length a units = 6$a^{2}$ sq.units.

Length of the edge of a cube =$\sqrt{\frac{s}{6}}$ units.

Formula

Surface area of cube = 6$a^{2}$ sq.units.

Where a= length of one side

Volume of a cube of edge length a units = a

Volume of cube = $(side \ length)^{3}$

=>512 $cm^{3}$ = $(side)^{3} $

=>side = cube root of 512

since 512 $cm^{3}$ = 8 cm $\times$ 8 cm $\times$ 8 cm

cube root of 512 = 8

Hence the side length of the cube is 8 cm.

Now,

surface area of cube = 6 $(side)^{2} $ = 6 $(8 \ cm)^{2}$ = 6 X 64 $cm^{2}$ = 384 $cm^{2}$

Each edge of a cube is increased by 10%. Find the percentage increase in the surface area of the cube.

Let side of cube = x

surface area = 6 $x^{2}$

new side = x $\times$ $\frac{110}{100}$ = $(1.1 \ x)^{2}$

new surface area = 6 $\times$ $(1.1 \ x)^{2}$ = 6 $(1.21 \ x)^{2}$ = 7.26

increase in area = 7.26 $x^{2}$ – 6$x^{2}$ = 1.26 $x^{2}$

% increase = $\frac{1.26 \ x^{2}}{6x^{2}}$ $\times$ 100

the percentage increase in the surface area of the cube= 21%