In a regular quadrilateral pyramid $S A B C D$ with vertex $S$, the side length of the pyramid's base is $b$, and the height of the pyramid is $b \sqrt{2}$. A sphere inscribed in this pyramid touches the lateral face $S A D$ at point $K$. Find the area of the section of the pyramid passing through the edge $A B$ and point $K$.
Let $S H$ be the height of the given pyramid, $S M$ be the apothem lying in the face $A S D$. Then the point $K$ of tangency of the sphere with the plane of this face lies on the segment $S M, M K=M H=\frac{\frac{\hbar}{2}}{2}$ as segments of tangents drawn to the sphere from one point. From the right triangle $S M H$ we find that

$$
S M=\sqrt{M H^{2}+S H^{2}}=\sqrt{\frac{b^{2}}{4}+2 b^{2}}=\frac{3}{2} b
$$

Then

$$
S K=S M-M K=\frac{3}{2} b-\frac{b}{2}=b, \frac{S K}{M K}=2
$$

which means, $K$ is the point of intersection of the medians of triangle $A S D$ and the line $A K$ intersects the edge $S D$ at its midpoint $P$. The cutting plane passes through the line $A B$, parallel to the plane $C S D$, and intersects this plane along the line $P Q$ (point $Q$ on the edge $S C$), so the section $A P Q B$ is a trapezoid with bases $A B=b$ and $P Q=\frac{\bar{b}}{2}$. Let $T-$ be the orthogonal projection of point $P$ onto the plane of the base of the pyramid, and $P N$ be the height of the trapezoid $A P Q B$. Then point $T$ lies on the diagonal $B D$ of the square $A B C D, T$ is the midpoint of $D H$, and $T N \perp A D$ by the theorem of three perpendiculars, with $T N=\frac{3}{4} A D=\frac{3}{4} b$ and $P T=\frac{1}{2} S H=\frac{b \sqrt{2}}{2}$. From the right triangle $P T N$ we find that

$$
P N=\sqrt{P T^{2}+T N^{2}}=\sqrt{\left(\frac{b \sqrt{2}}{2}\right)^{2}+\left(\frac{3}{4} b\right)^{2}}=\frac{b \sqrt{17}}{4}
$$

Therefore,

$$
S_{A P Q B}=\frac{1}{2}(A B+P Q) \cdot P N=\frac{1}{2}\left(b+\frac{b}{2}\right) \cdot \frac{b \sqrt{17}}{4}=\frac{3 \hbar^{2} \sqrt{17}}{16}
$$

![](https://cdn.mathpix.com/cropped/2024_05_06_a98533106cc029c284d7g-44.jpg?height=177&width=2039&top_left_y=266&top_left_x=-2)

The sphere inscribed in the regular pyramid $A B C D$ touches the face $A D C$ at point $K$. A section is made through the side $A B$ of the base $A B C$ of the pyramid and point $K$. Find the area of this section if the side of the base of the pyramid is $b$, and the height of the pyramid is $b \sqrt{2}$.

## Answer

$\sqrt{73}$

![](https://cdn.mathpix.com/cropped/2024_05_06_a98533106cc029c284d7g-44.jpg?height=57&width=183&top_left_y=746&top_left_x=0)