Question

what is the approximate circumference of the circle?
what is the height of the parallelogram?
what would be the approximate length of the parallelogram if the base were completely straight?
what would make the estimation of the circle’s area more precise?
what is the area of the circle used to create the parallelogram-like shape to the nearest tenth of a square unit?

Explanation:

1. Approximate Circumference of the Circle

Step1: Recall Circumference Formula

The formula for the circumference of a circle is \( C = 2\pi r \) or \( C=\pi d \), where \( r \) is the radius and \( d \) is the diameter. From the diagram, the radius \( r = 3 \), so the diameter \( d = 2r = 6 \).

Step2: Calculate Circumference

Using \( C = \pi d \) (or \( C = 2\pi r \)) and approximating \( \pi \approx 3.14 \):
\( C \approx 2 \times 3.14 \times 3 = 18.84 \) (or \( C \approx 3.14 \times 6 = 18.84 \)).

Step1: Identify Relationship

When a circle is divided into sectors and rearranged into a parallelogram, the height of the parallelogram equals the radius of the circle.

Step2: Determine Height

From the diagram, the radius \( r = 3 \), so the height of the parallelogram is equal to the radius, \( 3 \).

Step1: Recall Relationship

When the circle’s sectors are rearranged into a parallelogram, the length of the base (when straight) is half the circumference of the circle (since the circumference is split into two equal parts for the top and bottom of the parallelogram).

Step2: Calculate Half-Circumference

From the first problem, the circumference \( C \approx 18.84 \). Half of this is \( \frac{C}{2} \approx \frac{18.84}{2} = 9.42 \).

Answer:

\( \approx 18.8 \) (or \( 18.84 \), depending on precision)

2. Height of the Parallelogram