Why are some coins Reuleaux triangles?

Solution 1:

The Blaschke-Lebesgue Theorem states that among all planar convex domains of given constant width $B$ the Reuleaux triangle has minimal area.$^\dagger$

The area of the Reuleaux triangle of unit width is $\frac{\pi - \sqrt{3}}{2} \approx 0.705$, which is approximately $90\%$ of the area of the disk of unit diameter. Therefore, if one needs to mint (convex) coins of a given constant width and thickness, using Reuleaux triangles allows one to use approximately $10\%$ less metal.

$\dagger$ Evans M. Harrell, A direct proof of a theorem of Blaschke and Lebesgue, September 2000.

Solution 2:

The reason to not have a circle is likely purely aesthetic.

Once you've decided to mint a coin which isn't a circle, it's pretty important to still make one of constant width so it doesn't get stuck in machines. Also, many machines use the width of the coins to sort them (see for instance this youtube video showing one in action). That's a lot easier to do if you only have a single width for each coin instead of a range.

Solution 3:

Could it be to help the blind. I know it's not a uk coin, but the 50p and 20p are septagons to aid the blind. The old 10p was a similar size to the 50p; and the of 5p was a similar size to the 20p, so they made them a different shape. Also the 2p was a similar size to the 10p as well, so the 10p have grooves around the edge and the 2p is smooth. Same for the 1p (smooth)and it's similar size to the 5p (groved).

To summarize two groups of coin are a similar size:

1p (smooth edges), 5p (grooved edges), 20p (septagon shape). And the same for 2p, 10p and 50p, respectively.

The designs were to aid the blind who could feel the coin to know what is was as the sizes were too similar to distinguish.