Historically, almost everything traded against the US dollar. So how do you price a pair like EUR/GBP that doesn’t involve the dollar at all? You build it from the pairs that do.
Deriving a cross rate
A cross rate is an exchange rate worked out from two other pairs that share a common currency — usually the dollar. To get EUR/GBP, take EUR/USD and GBP/USD (both quoted against the dollar) and divide:
EUR/GBP = EUR/USD ÷ GBP/USD
The shared dollar leg cancels out, leaving the euro priced in pounds. The diagram shows the triangle: EUR and GBP both connect through USD, and the dashed line is the cross you’ve constructed.
Triangular consistency
This creates a tight relationship between three pairs. If EUR/USD, GBP/USD, and EUR/GBP ever drifted out of line, you could trade around the triangle — dollars to euros to pounds and back to dollars — and pocket a risk-free profit. That’s triangular arbitrage, and the very act of doing it pushes the rates back into agreement.
Because computers hunt these gaps in milliseconds, the three pairs stay consistent essentially all the time. The opportunity exists only in theory — which is exactly why the relationship holds.
The takeaway
A cross rate prices two currencies via a shared third (e.g. EUR/GBP from EUR/USD and GBP/USD). Triangular arbitrage keeps the three pairs consistent — any gap is instantly traded away, which is why cross rates always line up. (This is education, not investment advice.)