In a regular triangular pyramid, the side of the base is 4 cm, the height is 6 cm. Find the surface area of the pyramid.

The surface area of ​​a regular triangular pyramid is defined as the sum of the areas of the base and the lateral surface: S = Sb + S side.

At the base is a regular triangle, its area is determined by the formula Sbase = a2√3 / 4, where a is the side of the base.

The lateral surface area is defined as the sum of the lateral face areas. In a regular triangular pyramid, the side faces are equal isosceles triangles, therefore S side = 3 * 0.5 * a * h, where a is the side of the base, h is the apothem.

We find Apothema as the hypotenuse of a right-angled triangle, in which the legs are the height of the pyramid and the radius of the circle inscribed in the base: h = √ (r2 + H2).

The radius of the circle inscribed in the base is determined by the formula: r = a / 2√3.

r = a / 2√3 = 4 / 2√3 = 2 / √3;

h = √ (r2 + H2) = √ (4/3 + 36) = √ (112/3) = 4√7 / √3;

Side = 3 * 0.5 * a * h = 3 * 4 * 4√7 / 2√3 = 24√7 / √3;

Sb = a2√3 / 4 = 16√3 / 4 = 4√3;

S = S main + S side = 4√3 + 24√7 / √3 ≈ 43.59 cm2.