Given a regular triangular pyramid. The side of the base is 6, the height is equal to the root of 13. Find: apothem.

Data: H – height of the pyramid (H = √13 units); a – base side (a = 6 units)

To find out the length of the apothem (hypotenuse) in the indicated regular triangular pyramid, we apply the Pythagorean theorem: h = √ (H (height) ^ 2 + r (radius of the circle inscribed in the base (equilateral triangle)) ^ 2) = √ (H2 + (a * √3 / 6) ^ 2).

Let’s calculate: h = √ (H ^ 2 + (a * √3 / 6) ^ 2) = √ (√132 + (6 * √3 / 6) ^ 2) = √ (13 + 3) = 4 units.

Answer: The length of the apothem in the indicated regular triangular pyramid is 4 units.