The problem provided involves a triangular pyramid with the following known values:

Given:

1. Base sides: $AB = 20 \, \text{cm}, AC = 12 \, \text{cm}, BC = 16 \, \text{cm}$.
2. Dihedral angles at the base: $60^\circ$.
3. A sphere inscribed in the pyramid: Find its radius $R_{\text{sphere}}$.

Key Steps to Solve:

1. Understanding the dihedral angle:

   • The dihedral angle $60^\circ$ at the base indicates symmetry around the base plane.

2. Steps in the handwritten solution:

   • Step 1: $R_{\text{sphere}} = OO_1$ — the distance from the pyramid's apex $S$ to the center $O$ of the sphere.
   • Step 2: The segment $K O_1$ is tied to the geometry of triangle $ABC$ with $O_1$ as the center of the incircle.
   • Step 3: $K O$ acts as a bisector in triangle $ABC$, and the angle $\angle OK O_1 = 30^\circ$.

Strategy:

1. Compute the inradius $r$ of the base triangle $ABC$ using the formula: $r = \frac{A}{s}$ where $A$ is the area of $\triangle ABC$, and $s$ is its semiperimeter.

2. Use the geometric relations involving the bisectors, sphere center, and pyramid symmetry to derive the radius $R_{\text{sphere}}$.

Would you like me to proceed with a detailed step-by-step calculation or clarify any part of the solution so far?

Related Questions:

1. How is the inradius of a triangle calculated?
2. How do you determine the semiperimeter of a triangle?
3. What is a dihedral angle in a pyramid?
4. How do you find the radius of a sphere inscribed in a tetrahedron?
5. How is the relationship between the incircle and bisectors used in geometry?

Tip:

Always start solving 3D geometry problems by simplifying the 2D base first. Triangles form the foundation for most polyhedral problems!