How do you find the surface area of a similar sphere?

To find the surface area of a similar sphere, use the ratio of the radii squared times the original surface area.

When dealing with similar spheres, their surface areas are proportional to the square of the ratio of their radii. This means if you know the surface area of one sphere and the ratio of the radii between the two spheres, you can easily find the surface area of the other sphere.

Let's say you have two spheres, Sphere A and Sphere B. If the radius of Sphere A is \( r_A \) and the radius of Sphere B is \( r_B \), and you know the surface area of Sphere A is \( SA_A \), you can find the surface area of Sphere B (\( SA_B \)) using the following steps:

1. Calculate the ratio of the radii: \( \frac{r_B}{r_A} \).
2. Square this ratio: \( \left(\frac{r_B}{r_A}\right)^2 \).
3. Multiply the squared ratio by the surface area of Sphere A: \( SA_B = \left(\frac{r_B}{r_A}\right)^2 \times SA_A \).

For example, if Sphere A has a radius of 3 cm and a surface area of 113.1 cm², and Sphere B has a radius of 6 cm, the ratio of the radii is \( \frac{6}{3} = 2 \). Squaring this ratio gives \( 2^2 = 4 \). Therefore, the surface area of Sphere B is \( 4 \times 113.1 \) cm², which equals 452.4 cm².

This method leverages the geometric property that surface area scales with the square of the radius, making it straightforward to find the surface area of similar spheres.