A cube box is 20cm * 20cm * 20 cm What is the surface area of the largest size sphere that can fit in this box ? Leave your answer in terms of pi.
The longest possible diagonal of the cube will pass through its opposite corners. By using the Pythagorean theorem, we can calculate this diagonal as follows:
d = √(20^2 + 20^2 + 20^2)
= √(400 + 400 + 400)
= √1200
= 20√3
The diagonal represents the diameter of the largest sphere that can fit inside the cube. Therefore, the radius of the sphere is half the diameter, which is:
r = (20√3)/2
= 10√3
The surface area of the sphere is given by the formula:
A = 4πr^2
Substituting the value of r, we get:
A = 4π(10√3)^2
= 4π(100 × 3)
= 4π(300)
= 1200π
Therefore, the surface area of the largest size sphere that can fit inside the cube is 1200π square cm.