If a circle's radius is halved, how does this affect its area?

If a circle's radius is halved, how does this affect its area?

• A. The area remains the same
• B. The area is halved
• C. The area is reduced to one-fourth
• D. The area doubles

Answer: (C)

The area of a circle is calculated using the formula \( A = \pi r^2 \), where \( r \) is the radius. If the radius is halved, it becomes \( \frac{r}{2} \). Substituting this new radius into the area formula gives: \[ A' = \pi \left( \frac{r}{2} \right)^2 = \pi \left( \frac{r^2}{4} \right) = \frac{\pi r^2}{4} \] This shows that the new area \( A' \) is one-fourth of the original area \( A \). Thus, halving the radius reduces the area to one-fourth of its original size, confirming that the correct answer is indeed that the area is reduced to one-fourth. This understanding is crucial in various applications of geometry, particularly in aviation maintenance where calculations involving areas of components or materials are often necessary.

The area of a circle is calculated using the formula ( A = \pi r^2 ), where ( r ) is the radius. If the radius is halved, it becomes ( \frac{r}{2} ). Substituting this new radius into the area formula gives:

[ A' = \pi \left( \frac{r}{2} \right)^2 = \pi \left( \frac{r^2}{4} \right) = \frac{\pi r^2}{4} ]

This shows that the new area ( A' ) is one-fourth of the original area ( A ). Thus, halving the radius reduces the area to one-fourth of its original size, confirming that the correct answer is indeed that the area is reduced to one-fourth. This understanding is crucial in various applications of geometry, particularly in aviation maintenance where calculations involving areas of components or materials are often necessary.