Find the side surface and volume of the prism, whose base is a rhombus with a side equal to 6 cm

| March 30, 2021 | education

Find the side surface and volume of the prism, whose base is a rhombus with a side equal to 6 cm and an angle of 60 degrees, and the smaller diagonal makes angles of 45 degrees with the base and the side edge.

The smaller diagonal of the rhombus forms an equilateral triangle with the sides of the rhombus, since the angle between the sides is 60 °, which means it is 8 cm.

The smaller diagonal of the prism makes an angle of 45 ° with the base, but since the prism is straight, the other angle will be: 180 ° – (90 ° + 45 °) = 45 °, hence we have that the height of the prism is equal to the length of the smaller diagonal of the rhombus, that is 8 cm.

Perimeter of the base of the prism: P = 4a = 4 * 8 = 32.

Lateral surface of the prism: Sside = P * h = 32 * 8 = 256 cm2.

The area of ​​the base of the prism is found by the formula:

S = a ^ 2 * sin 60 °, where a is the side of the rhombus.

S = a ^ 2 * sin 60 ° = 64 * √3 / 2 = 32√3 m2.

The volume of the pyramid is: V = S * h. V = a ^ 2 * sin60 ° * h = 82 * √3 / 2 * 8 = 83 * √3 / 2 = 256 √3 cm3.

Answer: S side = 256 cm2., V = 256 √3 cm3.



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