Why is the half dollar coin bigger than a dollar coin?

Dollar coins that are gold colored are smaller than silver colored half dollars because the ones that were really gold were always smaller than the silver halves. Silver dollars that were really silver were bigger than the halves.

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At one time US Coins contained roughly their face value in silver, so their sizes were proportional. A quarter had about 25¢ worth of silver, a half contained 50¢ worth so it was twice as big, and a dollar was 4 times as big.

When silver was removed from coins in the 1960s there was no longer a need to keep sizes proportional because there was no longer any precious metal in coins. However the same sizes were kept so that millions of vending machines and coin sorters didn't have to be retrofitted.

On the other hand dollar coins hadn't been made since 1935 so there wasn't any problem with vending machines. When the Mint started making new dollars in 1971 they first tried to keep the old size, but the coins were too big to be practical. In 1979 a new smaller 26.5 mm size was adopted.

Other countries that have adopted completely new coinage systems no longer use just sizes to tell coins apart. For example in Europe, coins are made of different metals or even have two separate metals in them so that each one is distinctive regardless of size.