The diagonal drawn across ONE FACE of a cube measures 10 cm. Find the total surface area of the cube in ${\text{cm}}^2$.

A face is a square, so its diagonal gives $a^2+a^2=10^2$, that is $2a^2=100$ and $a^2=50{\text{cm}}^2$. The six faces then total $6\times 50=300{\text{cm}}^2$. The edge itself, √50 ≈ 7.07 cm, is never needed. (Reading 10 cm as the edge gives $600{\text{cm}}^2$; using the SPACE diagonal rule $3a^2=100$ — the diagonal from corner to opposite corner through the middle, not across a face — gives $a^2=33.33$ and $200{\text{cm}}^2$; taking $a=5$ cm gives $150{\text{cm}}^2.$)