Given a regular triangular prism, the perimeter of the base is 12 cm

Given a regular triangular prism, the perimeter of the base is 12 cm, the diagonal of the side face is 5 cm. Find the area of the lateral surface of the prism.

An equilateral triangle lies at the base of a regular triangular prism. Knowing the perimeter of the base of the prism, We can find the side of the base.

P = AB + BC + AC; AB = BC = AC = P / 3 = 12/3 = 4 (cm)

Knowing the diagonal of the lateral side, AB1 = 5 cm, and knowing the side of the base AB = 4 cm, we can find the height of the prism BB1 by applying the Pythagorean theorem to the ABB1 triangle: The square of the hypotenuse is equal to the sum of the squares of the legs.

AB1 ^ 2 = AB ^ 2 + BB1 ^ 2;

BB1 ^ 2 = AB1 ^ 2 – AB ^ 2;

BB1 ^ 1 = 5 ^ 2 – 4 ^ 2 = 25 – 16 = 9;

BB1 = √9 = 3 (cm).

Let’s find the area of ​​the side side АА1В1В. S = AB * BB1; S = 4 * 3 = 12 (cm ^ 2) /

The area of ​​the lateral surface of the prism will be equal to the sum of the areas of the lateral faces, and since our prism is correct, then the areas of the lateral faces are equal. We have 3 of them: АА1В1В, АА1С1С, ВВ1С1С.

S = 3 * S AA1B1B = 3 * 12 = 36 (cm ^ 2).

Answer. 36 cm ^ 2.