There are formulas for the area of triangles, various quadrilaterals, regular polygons, sections of circles and spheres (including whole circles and spheres). In general, you make use of the appropriate formula or combination of formulas for the shape of interest.
If the shape does not conform to one of these, but is described by a formula, then methods of calculus (integration) can be used to find the surface area.
A few of the relevant formulas are
.. triangle area = (1/2)bh . . . . b = base length; h = height measured perpendicular to the base
.. rectangle or parallelogram area = bh . . . . b and h defined as for a triangle
.. trapezoid area = (1/2)(b₁ +b₂)h . . . . b₁ and b₂ are the lengths of the parallel bases, h is the perpendicular distance between them
.. circle area = π*r^2 . . . . r is the radius
.. sphere area = 4π*r^2 . . . . r is the radius
.. lateral area of a cone = π*rh . . . . r is the radius of the base, h is the slant height
