The flag of the British Indian Ocean Territory (BIOT) features a blue ensign design with a Union Jack in the upper hoist-side quadrant and the BIOT coat of arms on the fly side. The aspect ratio of the flag is 1:2.

Colors used in the flag are:

1. Blue (Pantone 280 C): HEX #012169
2. Red (Pantone 186 C): HEX #C8102E
3. White: HEX #FFFFFF

The Union Jack occupies the top left corner, while the BIOT coat of arms is placed in the right half of the flag. The coat of arms consists of a white shield with wavy blue and white lines (representing the ocean), a palm tree, and a crown above the shield. Below the shield, there's a scroll with the territory's motto "In Tutela Nostra Limuria" written on it.

As for geometric properties and measurements, they can vary depending on the specific flag size being produced. However, the placement of the Union Jack and the coat of arms are consistent across all flag sizes.