A rectangular triangle with a hypotenuse of 8cm and an acute angle of 30 degrees rotates around

A rectangular triangle with a hypotenuse of 8cm and an acute angle of 30 degrees rotates around the smaller leg. calculate the surface area and volume of the body of revolution.

As a result of the rotation of this right-angled triangle around the smaller leg, a cone is obtained, the generatrix of which is equal to the hypotenuse of the triangle, and the height is equal to the smaller leg.

Since the smaller leg is opposite an angle of 30 °, its length is equal to half the length of the hypotenuse, that is, 8: 2 = 4 cm.

Using the Pythagorean theorem, we find the second leg:

x² + 4² = 8²,

x² = 64 – 16,

x = √48 (cm).

The length of this leg is equal to the radius of the base of the resulting cone.

Find the surface area of ​​the cone.

S = ∏ * 8 * √48 + ∏ * (√48) ² = ∏ * (32√3 + 48) =

16 * ∏ * (2√3 + 3) cm².

Let’s find the volume of the cone:

V = 1/3 * ∏ * (√48) ² * 4 = 64 * ∏ cm³.