# In a regular triangular prism ABCA1B1C1, the base area is 9, and the lateral edge is 4.

In a regular triangular prism ABCA1B1C1, the base area is 9, and the lateral edge is 4. Find the volume of the pyramid BACC1A1.

Since the pyramid is regular, the ABC triangle is equilateral.

Let us define the side of the triangle ABC through the area of ​​the base of the prism. Sbn = АС^2 * √3 / 4.

AC ^ 2 = 4 * Sb / √3 = 4 * 9 / √3 = 36 / √3 cm.

AC = 6 / √√3) .

The pyramid BAСС1A1 is a rectangular pyramid with the base AA1C1C, which has its height ВН.

The BH segment in an equilateral triangle ABC is its height, median and bisector, then BH = √3 * AC / 2 = √3 * 6 / √√3 / 2 = 3 * √ (√3).

Let’s define the area of ​​the rectangle АА1С1С. S = AC * AA1 = (6 / √√3) * 4 = 24 / √√3.

Then Vpyr = S * ВН / 3 = (24 / √√3) * 3 * √√3 / 3 = 24 / √3 = 8 * √3 cm3.

Answer: The volume of the pyramid is 8 * √3 cm3.

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