SUMMARY
The radius of a sphere inscribed in a square pyramid with a base edge of 10 and a slant height of 10 can be calculated using trigonometric principles. The cross-section of the pyramid forms an equilateral triangle, allowing the use of the tangent function. Specifically, the radius \( r \) is determined by the equation \( r = 5 \tan(30^{\circ}) \), resulting in \( r = \frac{5}{\sqrt{3}} = \frac{5}{3}\sqrt{3} \).
PREREQUISITES
- Understanding of basic geometry principles, specifically properties of pyramids.
- Familiarity with trigonometric functions, particularly tangent.
- Ability to interpret geometric diagrams and cross-sections.
- Knowledge of equilateral triangles and their properties.
NEXT STEPS
- Study the properties of inscribed spheres in polyhedra.
- Learn about the derivation of trigonometric functions in geometric contexts.
- Explore advanced geometric constructions involving pyramids and spheres.
- Investigate applications of geometry in real-world scenarios, such as architecture.
USEFUL FOR
Students studying geometry, educators teaching geometric principles, and anyone interested in solving problems related to inscribed shapes in three-dimensional figures.
