A circle with a 12 inch diameter is folded in half and then folded in half again. What is the area of the resulting shape?
Understand the Problem
The question is asking us to calculate the area of a shape that results from folding a circle with a 12 inch diameter in half twice. Initially, we find the area of the circle, and then we consider how folding affects the area.
Answer
$113.1$ square inches
Answer for screen readers
The area of the resulting shape after folding the circle twice is approximately $113.1$ square inches.
Steps to Solve
- Calculate the area of the original circle
The area (A) of a circle is calculated using the formula: $$ A = \pi r^2 $$ First, we find the radius. The diameter of the circle is 12 inches, so the radius (r) is: $$ r = \frac{12}{2} = 6 \text{ inches} $$ Now we can find the area: $$ A = \pi (6)^2 = \pi \times 36 \approx 113.1 \text{ square inches} $$
- Consider the effects of the first fold
When the circle is folded in half for the first time, the area of the resulting shape remains the same, which is approximately 113.1 square inches.
- Consider the effects of the second fold
Folding the shape in half again does not change the area either; it only changes its dimensions. Therefore, the area remains the same: $$ \text{Area after second fold} = 113.1 \text{ square inches} $$
The area of the resulting shape after folding the circle twice is approximately $113.1$ square inches.
More Information
This question illustrates that folding a shape does not change its area, only its form. The area is solely dependent on the original dimensions of the shape before any modifications.
Tips
- Misunderstanding the area change: Some might think that folding the shape in half changes the area, but it does not. Always remember that the area remains the same regardless of how many folds are made.
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