Two similar circles have their radii in the ratio 5:8. What is the ratio of their areas?

The correct option is 25:64.

To find the ratio of the areas of two similar circles when the ratio of their radii is given, we can use the fundamental geometric property relating linear dimensions to areas.

Step 1: Understand the formula for the area of a circle.
The area $A$ of a circle with radius $r$ is given by the formula:

$A=\pi r^2$

Step 2: Set up the ratio of radii.
Let the radii of the two circles be $r_1$ and $r_2$.
We are given that:

$\frac{r_1}{r_2}=\frac{5}{8}$

Step 3: Calculate the ratio of their areas.
Let the areas of the two circles be $A_1$ and $A_2$.
The ratio of their areas is:

$\frac{A_1}{A_2}=\frac{\pi {\left(r_1\right)}^2}{\pi {\left(r_2\right)}^2}$

$\frac{A_1}{A_2}={\left(\frac{r_1}{r_2}\right)}^2$

Step 4: Substitute the given ratio into the area ratio equation.

$\frac{A_1}{A_2}={\left(\frac{5}{8}\right)}^2=\frac{5^2}{8^2}=\frac{25}{64}$

Therefore, the ratio of their areas is 25:64.