An Ant's Journey

Geometry Level 5

There is a $4 \times 6 \times 18$ inch cereal box lying on the floor. It was placed face down on one of its $6 \times 18$ inch face. An ant is located at one of the bottom corners of the box. It wants to reach the opposite bottom corner (of the $6 \times 18$ face on the floor), by crawling along the outside of the box, or on the floor. The length of the shortest such path is $R$ inches. What is the value of $R^2$ ?

Details and assumptions

The ant may not crawl under the box.