78. Calculate the total length of the Reuleaux triangle and the area it encloses. Which is larger: the area enclosed by a circle or the area enclosed by a Reuleaux triangle of the same width? Also determine the measure of the internal angles at the vertices of the Reuleaux triangle.
78. If the width of the Reuleaux triangle (Fig. 252) is $h$, then each of the three arcs has a length of $\frac{2 \pi h}{6}$, and the length of the entire curve is $\pi h$. The area of the Reuleaux triangle is equal to the area of three circular sectors of radius $h$ with a central angle of $60^{\circ}$ minus twice the area of triangle $A B C$, i.e., it is

$$
\frac{h^{2}}{2}(\pi-\sqrt{3}) \approx 0.7048 h^{2}
$$

A circle of width $h$ has the same length $\pi h$ and a larger area $\frac{\pi h^{2}}{4} \approx 0.7854 h^{2}$.

Each arc connecting the vertices of the triangle contains $60^{\circ}$. Therefore, the angle between the tangent $A M$ and the chord $A B$ is $30^{\circ}$. The angle $CAN$ is also $30^{\circ}$ (see Fig. 252). Hence, the angle $M A N$ at the angular point $A$ of the Reuleaux triangle is $120^{\circ}$.