The ratio radii of two circles is 1:3. The ratio of their area is

The area of a circle is given by the formula:

\[
A = \pi r^2
\]

If the ratio of the radii of two circles is 1:3, let the radii of the smaller and larger circles be \( r_1 \) and \( r_2 \), respectively. Thus:

\[
\frac{r_1}{r_2} = \frac{1}{3}
\]

To find the ratio of their areas, we use the area formula:

\[
\frac{A_1}{A_2} = \frac{\pi r_1^2}{\pi r_2^2}
\]

Simplifying this, we get:

\[
\frac{A_1}{A_2} = \frac{r_1^2}{r_2^2} = \left( \frac{1}{3} \right)^2 = \frac{1}{9}
\]

Therefore, the ratio of the areas of the two circles is:

\[
1:9
\]

Would you like any further details or clarifications on this?

Here are five related questions:
1. How does the ratio of the diameters relate to the ratio of the areas?
2. If the circumference ratio is also 1:3, what does that imply about the relationship between circumference and radius?
3. How would the area ratio change if the radii ratio were 2:3 instead of 1:3?
4. What is the formula for the circumference of a circle, and how does it relate to the radius?
5. Can the area formula be applied in 3D objects, and if so, how?

**Tip:** In any geometric shape, areas scale by the square of the ratio of linear dimensions like radius or side length.

Explore how the ratio of the radii of two circles affects their area. This problem demonstrates key geometric concepts, including how the area scales with the square of the radius, using the formula A = πr^2. Suitable for students in Grades 6-8.

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