Here is a circle and a scaled copy of the circle with a scale factor of 2.
How does the circumference of the scaled copy compare to the circumference of the original circle?
The Correct Answer and Explanation is :
The image shows two concentric circles: a smaller one in blue and a larger one in green. Based on the description, the larger circle is a scaled copy of the smaller one with a scale factor of 2.
The formula for the circumference of a circle is:
[
C = 2\pi r
]
Where:
- ( C ) is the circumference.
- ( r ) is the radius of the circle.
In the original circle, the radius is ( r ), and the circumference is:
[
C_{\text{original}} = 2\pi r
]
When the circle is scaled by a factor of 2, the radius of the scaled circle becomes ( 2r ). The new circumference is:
[
C_{\text{scaled}} = 2\pi (2r) = 4\pi r
]
Thus, the circumference of the scaled copy is ( 4\pi r ), which is twice the original circumference. Therefore, the circumference of the scaled circle is doubled compared to the original circle.
Detailed Explanation
Scaling a geometric shape by a factor ( k ) increases all linear dimensions by ( k ). In the case of a circle, the radius, which is a linear dimension, scales by ( k = 2 ). Since the circumference of a circle is directly proportional to its radius, doubling the radius also doubles the circumference. This means the new circle has a circumference twice that of the original circle.
Although the area of a circle grows by the square of the scale factor (i.e., ( k^2 )), the circumference scales linearly with the radius. Thus, when the radius is doubled, the circumference also doubles.