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Problem : Perimeter is a measurement in how many dimensions?

One dimension

Problem : Why doesn't the shape of an object have an impact on its perimeter?

Because perimeter is a measure of length only; it doesn't concern direction. It measures length regardless of which direction a segment or curve is pointing, thus, shape doesn't affect perimeter.

Problem : What is the perimeter of a parallelogram with adjacent sides each of measure 8 inches? What is another name for this parallelogram?

32 inches. A rhombus.

Problem : If a regular polygon has an even integer for the lengths of its sides, is it possible to have an odd perimeter? Why or why not?

No, it's not possible. The perimeter of a regular polygon is ns, where n is the number of sides and s is the side length. Here the side length is even, so no matter how many sides the regular polygon has, it must have an even perimeter.

Problem : Two pentagons each have a perimeter of 23. Are their sides congruent?

Not necessarily. It's quite possible that the pentagons have all five sides of different length, and their sums are only equal by coincidence.