How many diagonals does a cube have?

Question

A cube has 8 corners and 12 edges. A diagonal is a straight segment joining two corners that are not joined by an edge. 1. How many diagonals does a cube have? 2. How many of them lie on a face, and how many pass through the interior? 3. If the cube has side 1, how long is each kind?

Accepted Answer

Sixteen diagonals: twelve face diagonals of length root 2 and four space diagonals of length root 3. All segments between corners. Eight corners give 8 choose 2 = 28 segments. Twelve of them are edges. The remaining 28 minus 12 = 16 are diagonals. Face diagonals. Each of the 6 faces is a square with 2 diagonals: 12 face diagonals. Each has length the square root of (1 + 1) = root 2 for a unit cube. Space diagonals. The other 16 minus 12 = 4 join opposite corners through the centre of the cube. Each has length the square root of (1 + 1 + 1) = root 3, about 1.732. The four space diagonals all p…

How many diagonals does a cube have?

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