19) A right pyramid with a square base has all edges of unit length. Its volume is
(A) $\frac{1}{3}$
(B) $\frac{\sqrt{2}}{6}$
(C) $\frac{\sqrt{3}}{6}$
(D) $\frac{\sqrt{2}}{2}$
(E) $\frac{\sqrt{3}}{3}$.
19) The answer is (B)

The apothem of the pyramid is equal to the height of an equilateral triangle with a unit side and is therefore $\frac{\sqrt{3}}{2}$. The height $h$ of the pyramid can be calculated by applying the Pythagorean theorem to a right triangle that has legs $h$ and half the base side, and hypotenuse the apothem;

thus $h=\sqrt{\left(\frac{\sqrt{3}}{2}\right)^{2}-\left(\frac{1}{2}\right)^{2}}=\sqrt{\frac{3}{4}-\frac{1}{4}}=\frac{1}{\sqrt{2}}$. The volume of the pyramid is therefore $\frac{1}{3} \cdot 1 \cdot 1 \cdot \frac{1}{\sqrt{2}}$ $=\frac{\sqrt{2}}{6}$.