The height of a regular quadrangular pyramid is 4. The side edge is 5. Find the diagonal of the base of the pyramid.

Given: regular quadrangular pyramid SABCD, height SO = 4 cm, SA = 5 cm.
Find: АС.
The SOA triangle is rectangular. Behind the Pythagorean theorem:
SA ^ 2 = SO ^ 2 + AO ^ 2 => AO ^ 2 = SA ^ 2 – SO ^ 2. We have:
AO ^ 2 = 5 ^ 2 – 4 ^ 2 = 25 – 16 = 9 (cm).
AO = Sqrt9 = 3 (cm).
Since the height of the pyramid divides its diagonal in half, AC = AO * 2. We have:
AC = 3 * 2 = 6 (cm).
Answer: AC = 6 cm.