The base perimeter of a regular triangular prism is 24 cm. Calculate the diagonal of the side

The base perimeter of a regular triangular prism is 24 cm. Calculate the diagonal of the side face if its area is 48 cm ^ 2.

A regular triangular prism is a prism with a regular triangle at its base. A regular triangle is a triangle in which all sides are equal, that is, an equilateral triangle.

The perimeter of a triangle is equal to the sum of the lengths of all its sides. To find the side of the base of the prism (or the side of the triangle), divide the perimeter by 3 (since 3 equal sides). a = P / 3.

a = 24: 3 = 8 (cm).

All three side faces of a regular triangular prism are equal, respectively, and their areas are equal. The side face is a rectangle in which one side is equal to the side of the base (8 cm), and the other is the height of the prism (it is also the side edge). The area of ​​a rectangle is equal to the product of its sides. S = ab. b = S / a.

b = 48: 8 = 6 (cm).

The two sides of the side face and its diagonal form a right-angled triangle. To find the diagonal, apply the Pythagorean theorem: The square of the hypotenuse is equal to the sum of the squares of the legs. x ^ 2 = a ^ 2 + b ^ 2.

x ^ 2 = 8 ^ 2 + 6 ^ 2;

x ^ 2 = 64 + 36;

x ^ 2 = 100;

x = 10 (cm).

Answer. 10 cm.