To understand the justification for the formula for the area of a circle, let's carefully break down the argument step-by-step.
1. The circumference [tex]\( C \)[/tex] of a circle is given by the formula [tex]\( C = 2\pi r \)[/tex], where [tex]\( r \)[/tex] is the radius. This tells us the total length around the circle.
2. Imagine dividing the circle into several equally-sized sectors (like slices of a pie). These sectors can be rearranged to form a shape that approximates a parallelogram.
3. When arranged into a parallelogram, the base of this parallelogram is half the circumference of the circle. Therefore, the base of the parallelogram is [tex]\( \pi r \)[/tex] (since [tex]\( \frac{1}{2} \times 2\pi r = \pi r \)[/tex]).
4. Now, considering what the height of this parallelogram would be.
In a circle, if you lay out the sectors, the height of the resulting parallelogram is essentially the same as the radius [tex]\( r \)[/tex] of the circle.
5. The area of a parallelogram is given by the product of its base and height. Thus, the area is calculated as:
[tex]\[ \text{Area} = \text{base} \times \text{height} = \pi r \times r = \pi r^2 \][/tex]
Therefore, with the word "radius" filling in the blank, the justification becomes clear: the height of the parallelogram is the radius [tex]\( r \)[/tex].
So, the word that best completes the argument is:
A. [tex]\( r \)[/tex]
Thus, the area of a circle is given by the formula [tex]\(\pi r^2\)[/tex].
