Question

a cube box is 20 cm x 20 cm x 30 cm what is the surface area of the largest size sphere that can fit in this box? leave your answer in terms of pi

Answer 1

To determine the largest sphere that can fit inside a rectangular box, we need to find the diameter of the sphere, which will be equal to the smallest dimension of the box.

The dimensions of the box are:
- Length: 30 cm
- Width: 20 cm
- Height: 20 cm

The smallest dimension here is 20 cm. Therefore, the diameter of the largest sphere that can fit in the box is 20 cm.

The radius \( r \) of the sphere is half of the diameter:

\[
r = \frac{d}{2} = \frac{20 \, \text{cm}}{2} = 10 \, \text{cm}
\]

The formula for the surface area \( A \) of a sphere is given by:

\[
A = 4\pi r^2
\]

Substituting the radius into the formula:

\[
A = 4\pi (10 \, \text{cm})^2
\]
\[
A = 4\pi (100 \, \text{cm}^2)
\]
\[
A = 400\pi \, \text{cm}^2
\]

Thus, the surface area of the largest sphere that can fit in the box is

\[
\boxed{400\pi} \, \text{cm}^2
\]