Use the Pythagorean theorem to find the heights of regular square pyramids, regular triangular pyramids, and other figures.
📘 Example ①
A regular square pyramid has a square base with side length 6cm and a slant height of 5cm. Find its height h.
Answer:Distance to the center of the base = diagonal/2 = 3√2 cm. h² = 5² - (3√2)² = 25 - 18 = 7. h = √7 cm
A regular square pyramid has a square base with side length 6cm and a slant height of 5cm. Find its height h.
Answer:Distance to the center of the base = diagonal/2 = 3√2 cm. h² = 5² - (3√2)² = 25 - 18 = 7. h = √7 cm
💡 Key Point
- Height of a regular square pyramid: the length from the vertex to the center of the base
- The slant height and “half the diagonal of the base” are perpendicular
Practice Problem
- A regular square pyramid has a square base with side length 4cm and a slant height of 5cm. Find its height (you may leave the answer with √).
Answer and Explanation
- Answer:√7 cm. Distance to the center of the base=diagonal/2=2√2. h²=25-(2√2)²=25-8=17. h=√17 cm. (In fact, h²=5²-(2√2)²=25-8=17, so the answer is √17.)