Question

explore the relationship between the radius and area of a circle by following these steps. 4. the height of the parallelogram is r. 5. the formula for the circumference of the circle is 2πr. 6. the base of the parallelogram - like figure is half the circumference of the circle, or ( b=\frac{1}{2}(2pi r)=pi r ). therefore, the area of the figure will be ( a = ) dropdown options: ( pi r ), ( (0)(2pi r) ), ( (r)(pi r) ) (images: a circle with radius ( r ) divided into sectors, and a parallelogram - like figure made from the sectors with base labeled ( \frac{1}{2}(2pi r)=pi r ))

Explanation:

Step1: Recall area of parallelogram

The area of a parallelogram is given by the formula \( A = \text{base} \times \text{height} \).

Step2: Identify base and height

From the problem, the base \( b \) of the parallelogram - like figure is \( \pi r \) (derived as half the circumference of the circle), and the height (from step 4) is \( r \).

Step3: Calculate the area

Substitute the values of base and height into the area formula of the parallelogram. So, \( A=\text{base}\times\text{height}=(\pi r)\times(r)=\pi r\times r=\pi r^{2} \). But looking at the given options, we use the formula \( A = \text{base} \times \text{height} \), where base is \( \pi r \) and height is \( r \), so \( A=(r)\times(\pi r) \).

Answer:

\((r)(\pi r)\) (which simplifies to \(\pi r^{2}\), and the option corresponding to this calculation is \((r)(\pi r)\))