Integrate between the bounds.
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If the shape is circle then circumference. If any other form (non-symmetrical) then you can measure the area with integration from Calculus.
Yes, if it is bound by plane figures, just add the area of each plane figure. If it has a curved surface, divide it into many small pieces, to approximate the area with small rectangles or triangles, then add them up.
To estimate area enclosed between the x-axis and a curve on a certain bounded region you can use rectangles or parallelograms.
You find the surface area of a cube by using the formula 6e2. In this case, e=edge.
You get the area by using formulas. There is usually a specific formula to find the area of each shape. Some irregular shaps may not have a formula.