Practice using the Pythagorean theorem in plane, solid, and coordinate contexts.
📘 Example ① (Comprehensive)
Find the volume and surface area of a regular square pyramid with a square base of side length 4cm and height 3cm.
Answer:Volume = (1/3)×16×3 = 16cm³。
Slant height l：l² = 3² + 2² = 13. l = √13.
One lateral face = (1/2)×4×√13 = 2√13. Four lateral faces = 8√13.
Surface area = 16 + 8√13 = (16 + 8√13)cm²
Find the volume and surface area of a regular square pyramid with a square base of side length 4cm and height 3cm.
Answer:Volume = (1/3)×16×3 = 16cm³。
Slant height l：l² = 3² + 2² = 13. l = √13.
One lateral face = (1/2)×4×√13 = 2√13. Four lateral faces = 8√13.
Surface area = 16 + 8√13 = (16 + 8√13)cm²
💡 Key Point
- Volume and surface area use different “heights” (the pyramid’s height versus its slant height)
Practice Problem
- Find the diagonal of a rectangular prism with length 6cm, width 8cm, and height 5cm.
Answer and Explanation
- Answer:5√5 cm. √(6²+8²+5²)=√(36+64+25)=√125=5√5.