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F'arc'tion

At the corner of the cube circular arcs are drawn and the area enclosed shaded. What fraction of the surface area of the cube is shaded? Try working out the answer without recourse to pencil and paper.

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Plutarch's Boxes

According to Plutarch, the Greeks found all the rectangles with integer sides, whose areas are equal to their perimeters. Can you find them? What rectangular boxes, with integer sides, have their surface areas equal to their volumes?

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Take Ten

Is it possible to remove ten unit cubes from a 3 by 3 by 3 cube made from 27 unit cubes so that the surface area of the remaining solid is the same as the surface area of the original 3 by 3 by 3 cube?

Changing Areas, Changing Volumes

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2
Try working on Changing Areas, Changing Perimeters first, as it is a two-dimensional version of this three-dimensional problem.
 
If I know two rectangles have the same area, how can I decide, just by looking at their dimensions, which has the greater perimeter?
If I know two cuboids have the same volume, how can I decide, just by looking at their dimensions, which has the greater surface area?