The base of the straight prism is a triangle with sides of 5 cm and 3 cm and an angle of 120 ° between them.

The base of the straight prism is a triangle with sides of 5 cm and 3 cm and an angle of 120 ° between them. The largest of the areas of the side faces is 56 cm ^ 2. Find the total surface area of the prism.

By the cosine theorem, we find the third side of the base:

c ^ 2 = a ^ 2 + b ^ 2 – 2 * a * b * cos 120 ° = 5 ^ 2 + 3 ^ 2 – 2 * 5 * 3 * (- 0.5) = 25 + 9 + 15 = 49 = 72;

c = 7 cm.

The base area is determined by the formula:

Sbas = 0.5 * a * b * sin 120 ° = 0.5 * 5 * 3 * √3 / 2 = 15√3 / 4 cm2.

Obviously, the largest area of ​​the side faces is that face, one of the sides of which is the larger side of the base. The second side of this face is the height, knowing the area and side of the base, we find the height of the prism:

h = S / c = 56/7 = 8 cm.

The area of ​​the lateral surface of a straight prism is equal to the product of the height of the prism and the perimeter of its base:

Sside = P * h = (5 + 3 + 7) * 8 = 15 * 8 = 120 cm2.

The total surface area is equal to the sum of the lateral surface areas and the two bases:

S total = S side + 2 * Sb = 120 + 2 * 15√3 / 4 = 120 + 15√3 / 2 ≈ 133 cm2.