Circle Area Ratio Explained

This solution details how to calculate the ratio of the areas of two circles when the ratio of their radii is provided.

Problem: Circle Radii to Area Ratio

The question states that the ratio of the radii of two circles is $1 : 4$. We need to determine the ratio of their corresponding areas.

Mathematical Foundation

• The area of a circle is given by the formula $A = \pi r^2$.
• The ratio of areas depends directly on the square of the ratio of their radii.

Area Ratio Calculation Steps

1. Define radii and areas: Let the radii of the two circles be $r_1$ and $r_2$, and their areas be $A_1$ and $A_2$.
2. Given radii ratio: We are given $\frac{r_1}{r_2} = \frac{1}{4}$.
3. Area formula: The areas are $A_1 = \pi r_1^2$ and $A_2 = \pi r_2^2$.
4. Calculate area ratio: The ratio of the areas is $\frac{A_1}{A_2} = \frac{\pi r_1^2}{\pi r_2^2}$.
5. Simplify the ratio: The $\pi$ cancels out, leaving $\frac{A_1}{A_2} = \frac{r_1^2}{r_2^2} = \left(\frac{r_1}{r_2}\right)^2$.
6. Substitute the radii ratio: Substitute the given value: $\frac{A_1}{A_2} = \left(\frac{1}{4}\right)^2$.
7. Final calculation: Squaring the fraction gives $\frac{1^2}{4^2} = \frac{1}{16}$.

Thus, the ratio of the areas is $1 : 16$.

Conclusion

The ratio of the areas of the two circles is $1 : 16$. This corresponds to Option 4.