Area is a vector quantity...why???

Yes, It is a Vector quantity,If you are dealing with physics problems where a volume is enclosed by a  closed surface. Especially when we are having a vector field like electric field  and magnetic field enclosed in a closed surface. we define a quantity called flux which is related to number of lines of force of the field passing through the surface area. we have to consider area as a vector quantity as we change the angle or orientation of the surface area the number of lines of force change significantly. It is possible that the lines of force crossing the area become zero in some orientation. so to include the effect of angle and orientation from a perticular reference we need to consider area as vector quantity. We can look it up in Gauss divergence theorem.

When we draw a small rectangle on a graph paper having x-y coordinates, we take the product length x breadth as area. Now we are not bothered about the location of the plane, since we know that it is on the x-y plane. But if we draw the same rectangle,  on a 3 dimensional/ cylindrical coordinate system we have to be specific regarding the location of the plane (where the rectangle is drawn) with respect to the origin of the cordinate system. The rectangle which we are concerned can be drawn on any plane such as an inclined plane, vertical plane or horizontal plane. In order to get the information about the orientation of a plane, we have to specify the area as a vector quantity . Hence  area is specified as a vector quantity.

Yes, scalar and Yes, vector depending on the situation.

CASE-1

Let us take example for SCALAR:

A human being wants to find out area of his farm. Then, area is scalar.

But, why NOT VECTOR?

It is because of the logic that we define things depending on the context. Here, why would you require direction of an area. So, here we define area as a SCALAR quantity.

CASE II: VECTOR

In electromagnetism, we often require the direction of a loop, say if a conducting wire on a surface has current flowing through it in CLOCKWISE direction then it has area VECTOR pointing inwards i. e. into the plane of paper. Whereas, when we flip the wire so that the current flows in ANTI CLOCKWISE direction, then the area VECTOR points outwards i. e. outside plane of the paper. In this case, area is a vector.

Area is not a vector quantity. It is used as a vector quantity in many situations just for mathematical usage and the simplicity it provides.

The plane surface that is two-dimensional reduces to one-dimension by assigning a vector whose magnitude is equal to area of that plane and direction is perpendicular to all lines lying on that plane. Given this vector, one can construct back the plane. This is a useful concept in studying complex structures, in which we reduce all planes into vectors.

When we consider flow of something through a surface, called flux, we use area as a vector as it helps in defining flux as dot product of vector quantity that flows out with area vector. Here, area cannot be used as a scalar quantity, as for finding flux, orientation of area plays an important role. By changing orientation you change amount of flow through the area.

Area is a scalar quantity. Area is the product of two lengths. Length is a scalar quantity, direction in the length being measured does not influence the measurement. For example, length of a line drawn between two points A and B is the same for AB and BA. As the product of two scalar quantities is also a scalar quantity, area is a scalar quantity.

Commoly area is a scalar quantity because there is no need of direction to define and also follow the algebrac summation. When we talk about vector there exist a frame of reference with a certain origin. Dealing with physical quantity,  which depends upon area, we have to treat area as vector field which is known as areal vector. Still only area is a scalar.