Area of a Circle: How many times does the shape fit inside of another shape?

To solve this problem, we need to find how many times the area of the small circle fits into the area of the large circle. We'll do this by calculating both areas and finding their ratio.

Step 1: Identify the given information
We have two circles:

• Small circle with radius $r_1 = 4$ cm
• Large circle with radius $r_2 = 10$ cm

Step 2: Calculate the area of the small circle
Using the formula for the area of a circle $A = \pi r^2$, we get:
$A_1 = \pi \cdot 4^2 = \pi \cdot 16 = 16\pi$ square cm

Step 3: Calculate the area of the large circle
Similarly, for the large circle:
$A_2 = \pi \cdot 10^2 = \pi \cdot 100 = 100\pi$ square cm

Step 4: Find how many times the small area fits into the large area
We divide the large area by the small area:
$\frac{A_2}{A_1} = \frac{100\pi}{16\pi} = \frac{100}{16}$

Step 5: Simplify the fraction
$\frac{100}{16} = \frac{25}{4}$

Step 6: Convert to a mixed number
$\frac{25}{4} = 6\frac{1}{4}$

This makes sense because when we scale a circle's radius by a factor of $\frac{10}{4} = 2.5$, its area scales by the square of that factor: $(2.5)^2 = 6.25 = 6\frac{1}{4}$.

Therefore, the area of the small circle fits into the large circle $6\frac{1}{4}$ times.