$$h_1=\sqrt{{a_1}^2-{\left(\frac{a_1}{2}\right)}^2}=\sqrt{{a_1}^2-\frac{{a_1}^2}{4}}=\sqrt{\frac{4·{a_1}^2}{4}-\frac{{a_1}^2}{4}}=\sqrt{\frac{3·{a_1}^2}{4}}$$

$$h_1=\frac{\sqrt{3}·a_1}{2}$$

$$h_1=\frac{\sqrt{3}·10·\sqrt{3}}{2}=\frac{3·10}{2}$$

$$h_2=\frac{\sqrt{3}·a_2}{2}$$

$$h_2=\frac{\sqrt{3}·4·\sqrt{3}}{2}=\frac{3·4}{2}$$

$$H^2={\left(3\sqrt{7}\right)}^2-6^2=9·7-36=63-36$$

$$H^2=27=9·3$$

$$H=\sqrt{9·3}$$

$$B_1=\frac{a_1·h_1}{2}=\frac{10\sqrt{3}·15}{2}$$

$$B_2=\frac{a_2·h_2}{2}=\frac{4\sqrt{3}·6}{2}$$

$$V=\frac{H}{3}\left(B_1+\sqrt{B_1·B_2}+B_2\right)$$

$$V=\frac{\bcancel{3}\sqrt{3}}{\bcancel{3}}·\left(75\sqrt{3}+\sqrt{75\sqrt{3}·12\sqrt{3}}+12\sqrt{3}\right)$$

$$V=\sqrt{3}·\left(75\sqrt{3}+\sqrt{2700}+12\sqrt{3}\right)$$

$$V=\sqrt{3}·\left(75\sqrt{3}+\sqrt{900·3}+12\sqrt{3}\right)$$

$$V=\sqrt{3}·\left(75\sqrt{3}+30\sqrt{3}+12\sqrt{3}\right)$$

$$V=\sqrt{3}·\left(117\sqrt{3}\right)$$

$${h_3}^2=H^2+3^2$$

$${h_3}^2={\left(3\sqrt{3}\right)}^2+3^2$$

$${h_3}^2=27+9=36$$

$$h_3=\sqrt{36}$$

$$A=B_1+B_2+M$$

The lateral area consists of three isosceles trapezoids.

$$M=3·\frac{\left(a_1+a_2\right)}{2}·h_3$$

$$M=3·\frac{\left(10\sqrt{3}+4\sqrt{3}\right)}{2}·6$$

$$A=B_1+B_2+M=75\sqrt{3}+12\sqrt{3}+126\sqrt{3}$$