In a straight triangular prism, all edges are the same length. Find the volume of the prism if the side

In a straight triangular prism, all edges are the same length. Find the volume of the prism if the side surface area is 12 cm2.

The area of ​​the lateral surface of the prism is equal to the product of the area of ​​the lateral face by the number of these faces:

S side. p. = S side. gr. ∙ n = a ∙ l ∙ n, where a is the base edge, l is the lateral edge of the prism, n = 3 (triangular prism).

Since a = l, we find a:

S side. p. = a2 ∙ n;

12 = 3a ^ 2;

a ^ 2 = 4;

a = 2 (cm).

The volume of the straight prism is equal to the product of the base area by the lateral edge V = Sbase. ∙ l.

An equilateral triangle lies at the base of the prism, its area is found by Heron’s formula.

S = √ (p (p – a) (p – b) (p – c)), where p = (a + b + c) / 2.

p = (2 + 2 + 2) / 2 = 3 (cm).

S = √ (3 ∙ (3 – 2) (3 – 2) (3 – 2)) = √3 (cm2).

V = 3√3 (cm3).

Answer: V = 3√3 (cm3).