Question

question
the figure shows how the formula for the circumference of a circle may be used to derive the formula for the area of a circle.
$c = 2\pi r$
$a = ?$
based on the figure, which of the following represents the area of the circle?
standard(s)
nc.7.g.4.i: understand the relationships between the radius, diameter, circumference, and area.
other questions on this standard: 22
answer
a $a = \pi r \cdot r^{2}$
b $a = \pi r \cdot r$
c $a = 2\pi r \cdot r^{2}$
d $a = 2\pi r \cdot r$

Explanation:

Step1: Identify shape approximation

The circle is rearranged into a shape approximating a parallelogram.

Step2: Find base of parallelogram

The base is half the circle's circumference: $\frac{2\pi r}{2} = \pi r$

Step3: Find height of parallelogram

The height equals the circle's radius: $r$

Step4: Calculate area (parallelogram formula)

Area = base $\times$ height: $A = \pi r \times r$

Answer:

B. $A = \pi r \cdot r$