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

$$\boxed{{\displaystyle 400\pi }},{\text{cm}}^2$$