The body is obtained by rotating a rhombus with side a and an acute angle alpha around a straight

| April 28, 2021 | education

The body is obtained by rotating a rhombus with side a and an acute angle alpha around a straight line containing the diagonal of the rhombus. Find the volume of the resulting body of revolution.

There are two cones with a common base with a rhombus rotation around the AC axis.

The radius of these cones is the OВ segment, and its height is the OA segment.

From the right-angled triangle AOB, we determine the lengths of the legs OA and OB.

Since the triangle AOB is isosceles, AB = AD = a cm, then the height of AO is also its bisector, then the angle OAB = α / 2.

Cos (α / 2) = AO / AB.

AO = AB * Cos ((α / 2) = a * Cos (α / 2).

Sin (α / 2) = OB / AB.

OB = AB * Sin (α / 2) = a * Sin (α / 2).

Let’s define the volume of one cone.

Vfin = π * ОВ ^ 2 * AO / 3 = π * a ^ 2 * Sin2 (α / 2) * a * Cos (α / 2) / 3 = π * a ^ 3 * Sin2 (α / 2) * Cos (α / 2) / 3 cm3.

Then the volume of the rotation figure is: V = 2 * Vfin = 2 * π * a ^ 3 * Sin2 (α / 2) * Cos (α / 2) / 3 cm3.

Answer: The volume of the rotation figure is 2 * π * a ^ 3 * Sin2 (α / 2) * Cos (α / 2) / 3 cm3.



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