1. Dana is a regular hexagonal pyramid with the length of the base edge $a=10 \mathrm{~cm}$ and the length of the side edge $s=13 \mathrm{~cm}$.

a) Accurately calculate the surface area of the pyramid.

b) Accurately calculate the area of the axial section of the pyramid, which is formed when the pyramid is intersected by a plane passing through the apex of the pyramid and two vertices of the base.

(10 points)
1. 

a) Using the Pythagorean theorem, we calculate the height of the side face of the pyramid $v_{s}=12 \mathrm{~cm}$. Using the formula for calculating the area of the base face $S_{op}=\frac{6 a^{2} \sqrt{3}}{4}=\frac{6 \cdot 10^{2} \sqrt{3}}{4}=150 \sqrt{3}$ $\mathrm{cm}^{2}$ and the area of the mantle $S_{pl}=6 \cdot \frac{a \cdot v_{s}}{2}=6 \cdot \frac{10 \cdot 12}{2}=360 \mathrm{~cm}^{2}$, we calculate the surface area of the pyramid $P=(150 \sqrt{3}+360) \mathrm{cm}^{2}$.
Calculation of the height of the side face of the pyramid $v_{s}=12 \mathbf{cm}$ ..... 1 point
Writing or using the formula for calculating the area of the base face $S_{op}=\frac{6 a^{2} \sqrt{3}}{4}$ ..... 1 point
Calculation $S_{op}=150 \sqrt{3} \mathbf{cm}^{2}$ ..... 1 point
Writing or using the formula for calculating the area of the mantle $S_{pl}=6 \cdot \frac{a \cdot v_{s}}{2}$ ..... 1 point
Calculation $S_{pl}=360 \mathbf{cm}^{2}$ ..... 1 point
Solution $P=(150 \sqrt{3}+360) \mathbf{cm}^{2}$ ..... 1 point

b) We determine that the axial section of the pyramid is an isosceles triangle with sides $20 \mathrm{~cm}, 13 \mathrm{~cm}$, and 13 $\mathrm{cm}$. Using the Pythagorean theorem, we calculate the height of the triangle, which is the axial section of the pyramid $v_{op}=\sqrt{69} \mathrm{~cm}$. We calculate the area of the axial section $S_{op}=\frac{2 a \cdot v_{op}}{2}=\frac{20 \cdot \sqrt{69}}{2}=10 \sqrt{69} \mathrm{~cm}^{2}$.

Determination that the axial section of the pyramid is an isosceles triangle with sides $20 \mathbf{cm}, 13 \mathbf{cm}$, and $13 \mathbf{~cm}$ 1 point Writing or using the formula for calculating the height of the triangle, which is the axial section of the pyramid $v_{op}=\sqrt{69} \mathbf{~cm}$ 1 point Writing or using the formula for calculating the area of the axial section $S_{op}=\frac{2 a \cdot v_{op}}{2} \ldots .1$ point Solution $S_{op}=10 \sqrt{69} \mathbf{cm}^{2}$ 1 point