Polygon Bowl

Geometry Level 5

Four regular hexagons with side length $22$ are attached by their sides to a square, also with side length $22$ . The hexagons are then folded upwards to create a bowl. From base to rim, the height of this bowl can be expressed as $\sqrt{a}$ , where $a$ is a positive integer. Find the value of $a$ .

Details and assumptions

From base to rim is the height of the bowl normal to the base, i.e straight up and down.