## Task B-3.8.

A regular quadrilateral truncated pyramid whose base edges are $a=12 \mathrm{~cm}$ and $c=8 \mathrm{~cm}$, and all lateral edges are $b=20 \mathrm{~cm}$, is intersected by a plane passing through the endpoint of the diagonal of the smaller base, perpendicular to that diagonal. What is the surface area of the smaller part of the pyramid that is created by this intersection?
## Solution.

Sketch:

![](https://cdn.mathpix.com/cropped/2024_05_30_c3f5968a3447b3c0b4b2g-14.jpg?height=675&width=919&top_left_y=1774&top_left_x=576)

Introduce the following notations: $v$ is the height of the frustum of the pyramid, $d=|A C|, d_{1}=|E G|, x=|P K|, y=|P C|$.

The surface area of the resulting pyramid $K L C G$ (with vertex at $G$ and base $K L C$) is calculated as the sum of the areas:

$$
O=P_{\triangle K L G}+2 \cdot P_{\triangle K C G}+P_{\triangle K C L}
$$

To calculate the area of the given section, the triangle $\triangle K L G$, determine:

$$
\begin{aligned}
& y=\frac{d-d_{1}}{2}=\frac{12 \sqrt{2}-8 \sqrt{2}}{2}=2 \sqrt{2} \mathrm{~cm} \\
& v=\sqrt{b^{2}-y^{2}}=\sqrt{400-(2 \sqrt{2})^{2}}=\sqrt{392}=14 \sqrt{2} \mathrm{~cm}
\end{aligned}
$$

From the similarity of triangles $\triangle M B C$ and $\triangle P K C$, it follows that $\frac{\frac{d}{2}}{x}=\frac{\frac{d}{2}}{y}$, i.e., $x=y=2 \sqrt{2} \mathrm{~cm}$

and $\frac{|K C|}{a}=\frac{y}{\frac{d}{2}}$, i.e., $|K C|=4 \mathrm{~cm}$.

$$
P_{\triangle K L G}=\frac{2 x v}{2}=56 \mathrm{~cm}^{2}
$$

The base of the pyramid is $\triangle K L C$, and its area is

$$
P_{\triangle K L C}=\frac{2 x y}{2}=8 \mathrm{~cm}^{2}
$$

The lateral faces of the pyramid are congruent triangles $\triangle K C G$ and $\triangle L C G$ with a base of length $|K C|=4 \mathrm{~cm}$ and height equal to the height of the isosceles trapezoid $B C G F$.

Thus,

$$
P_{\triangle K C G}=\frac{4 \cdot \sqrt{400-4}}{2}=12 \sqrt{11} \mathrm{~cm}^{2}
$$

Finally, $O=56+8+2 \cdot 12 \sqrt{11}=64+24 \sqrt{11} \mathrm{~cm}^{2}$.

## SCHOOL/CITY COMPETITION IN MATHEMATICS

## 4th grade - high school - B category

February 4, 2010.

IF A STUDENT HAS A DIFFERENT METHOD OF SOLVING THE PROBLEM, THE COMMITTEE IS OBLIGED TO EVALUATE AND GRADE THAT METHOD APPROPRIATELY.