Final answer:
The geometrical shape with circumference, radius, and diameter is a circle. The area of a circle is given by πr² and the circumference is 2πr, which is dimensionally consistent as perimeters and areas are measured in lengths and square units respectively.
Step-by-step explanation:
The geometrical shapes that have a circumference, radius, and diameter are characteristics of a circle. The formula for the area of a circle is πr², while the circumference is 2πr. In a practical example, if we place a circle within a square where the side of the square is equal to the diameter of the circle (a = 2r), we can compare their perimeters and areas.
For the square, the perimeter would be 4a and the area a². For the circle fitting inside, its perimeter (circumference) would be approximately 3 times the side of the square (6r), given the estimation that perimeter is somewhat close to 2πr. The area of the circle would be less than that of the square but larger than half its area—closer to three-quarters. Thus, it can be approximated to 3r², which correctly simplifies to πr², the actual formula for the area of a circle.
When considering dimensional analysis, it's important to remember that perimeters are described in units of length, and areas in square units, which helps distinguish between formulas for circumference (length) and area (square).