The side of the base of a regular triangular prism is 4 cm, and the side edge is 2√3. how to find the volume?

1. Known rule of Heron for calculating the area of ​​a triangle on its three sides:

S triangle = {P * (p – a) * (p – b) * (p – c)} ^ 1/2 <where p = (a + b + c): 2.

2. The volume V of the prism is equal to the product of the area of ​​its base by the height.

3. Let’s calculate what is the area S of a regular triangle at the base of the prism, if each side of it is 4 cm.

To do this, we first find the semiperimeter p of the triangle.

p = (4 cm + 4 cm + 4 cm): 2 = 6 cm.

S = (6 * 8 * 8 * 8) ^ 1/2 = (48 * 8 ^ 2) ^ 1/2 = 8 * (16 * 3) ^ 1/2 = 8 * 4 * 3 ^ 1/2 =

32 * 3 ^ 1/2.

4. Let us calculate what the volume V of the prism is equal to if, according to the condition of the problem, the lateral edge H

equals 2 * 3 ^ 1/2 cm.

V = S * H ​​= 32 * 3 ^ 1/2 * 2 * 3 ^ 1/2 = 64 * 3 = 192 cm ^ 3.

Answer: The volume of the prism is 192 cubic centimeters.