Area of unshaded portion

Question

There is a cube. Each face of the cube has an arc starting from one corner to the opposite corner. The enclised area is shaded. What is the area of the unshaded area for all the 6 faces?

Answer 1

Hi Debyani,

 

If I understand the question correctly, the enclosed area by the arc, that is the larger portion a a given face, is shaded. 

 

Then, let the side length of the cube be x. This will also be equal to the radius of a circle centered on one of the vertices on a given face of the cube.

The area enclosed by the arc on a given face will be (call it S)

S = pi*x^2/4 (divide by 4 because the area is a quarter of the circle's area. Central angle = 90 deg)

the area of a given face of the cube is x^2

Then, the unshaded area on a given face is equal to

x^2 - pi*x^2/4

We need to multiply this by 6 to find it for the whole cube.

6*(x^2 - (pi*x^2)/4)

= 6x^2 - 3/2*pi*x^2)