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increase the surface area of a solid means to increase the area of solid
You increase the surface area
It really depends on what solid you refer to. Basically, the surface area is basically taking a shape's net and finding the area of that.
The "Surface Area" of the solid figure. Note, the word "total" in the answer above is not correct/needed - there can not be anything less than a surface area of a solid figure.
It will increase it, as more of the outer surface of the solid is incontact with the liquid that it is dissolving in. :)