Question
The radius of a circle is doubled. Which of the following describes the effect of this change on the area?
Ask by Barker Warner.
Mar 21,2025 02:25
UpStudy AI Solution
Tutor-Verified Answer
Answer
The area of the circle becomes four times larger.
Solution
1. The area \( A \) of a circle with radius \( r \) is given by:
\[
A = \pi r^2
\]
2. If the radius is doubled, the new radius becomes \( 2r \).
3. Substitute the new radius into the area formula:
\[
A_{\text{new}} = \pi (2r)^2
\]
4. Simplify the expression:
\[
A_{\text{new}} = \pi (4r^2) = 4\pi r^2
\]
5. Since the original area is \( \pi r^2 \), the new area is:
\[
A_{\text{new}} = 4A
\]
The area of the circle is quadrupled.
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Simplify this solution Beyond the Answer
When you double the radius of a circle, the area doesn’t just double—it quadruples! This happens because the area of a circle is given by the formula A = πr². So, if you change the radius from r to 2r, the new area becomes A = π(2r)² = π(4r²), which means you end up with four times the original area. Pretty cool, right?
This principle is not just applicable to circles; it appears in various geometric shapes! For instance, if you double the side length of a square, the area increases by a factor of four as well. Understanding how dimensions affect areas helps in fields like architecture and design, ensuring efficient use of space while maximizing aesthetics!
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