Final answer:
The provided formulas for surface area of a square, circle, triangle, and rectangle are essential for determining which shape can have a surface area of 90 square inches, but specific dimensions of each shape are needed to find an exact answer. Dimensional analysis ensures formulas are consistent with units of measurement.
Step-by-step explanation:
To find the shape with a surface area of 90 square inches in a geometric context, we would need more information about the specific dimensions of each shape to provide an exact answer. However, we can discuss each shape's surface area formula to understand how it could be determined. For example, a square with side length 'a' would have a surface area expressed by the formula A = a². A circle with radius 'r' would have a surface area found through the formula A = πr². A triangle, specifically an equilateral triangle with side length 'a', would have a surface area given by A = (√3/4)a². Lastly, a rectangle with length 'l' and width 'w' would have a surface area of A = lw.
Using dimensional analysis, we can ensure the formulas are dimensionally consistent by verifying that the units make sense; area should always be expressed as a unit squared. Moreover, we can relate the shapes to each other, such as an inscribed ciercle within a square, where the circle's area is less than the square but more than half of it. Understanding the formulas and how to apply them is important, but knowing exact numbers for each shape's dimensions is necessary to calculate the specific surface areas.