2.6. The center of the upper base of a cube with an edge equal to $a$ is connected to the midpoints of the sides of the lower base, which are also connected in sequential order. Calculate the total surface area of the resulting pyramid.
2.6. Since the edge of the cube is $a$, the side of the base of the pyramid $S A B C D$ is $\frac{a \sqrt{2}}{2}$ (Fig. 2.6). Considering that $O K=\frac{1}{2} A D=\frac{a \sqrt{2}}{4}$, we will find the apothem of the pyramid:

![](https://cdn.mathpix.com/cropped/2024_05_21_1e3d8d2f6605ac23e2a5g-458.jpg?height=541&width=456&top_left_y=1333&top_left_x=689)

Fig. 2.6

$$
S K=\sqrt{S O^{2}+O K^{2}}=\sqrt{a^{2}+\frac{a^{2}}{8}}=\frac{3 a \sqrt{2}}{4}
$$

Thus,

$$
S_{\text {side }}=4 \cdot \frac{1}{2} \cdot \frac{a \sqrt{2}}{2} \cdot \frac{3 a \sqrt{2}}{4}=\frac{3 a^{2}}{2}, S_{\text {total }}=\left(\frac{a \sqrt{2}}{2}\right)^{2}+S_{\text {side }}=2 a^{2}
$$

![](https://cdn.mathpix.com/cropped/2024_05_21_1e3d8d2f6605ac23e2a5g-459.jpg?height=523&width=421&top_left_y=475&top_left_x=47)

Answer: $2 a^{2}$.