A rectangle with a hypotenuse of 6 cm and an acute angle of 30 degrees rotates around the smaller leg.

A rectangle with a hypotenuse of 6 cm and an acute angle of 30 degrees rotates around the smaller leg. Find the surface area of the resulting body of revolution.

The result of the rotation of this right-angled triangle will be a cone, in which the smaller leg is the height of the cone, and the larger leg is the radius of the circumference of the base of the resulting cone, the hypotenuse of the triangle is the length of the generatrix of the cone.
Let’s find the legs of the given triangle.
The smaller leg is located opposite an angle of 30 °, which means that it is equal to half of the hypotenuse – 6/2 = 3 (cm).
We find the larger leg by the Pythagorean theorem:
√ (6² – 3²) = √ (36 – 9) = √27 = 3√3 (cm).
In our cone, we got:
h = 3 cm, r = 3√3 cm, l = 6 cm.
The condition does not indicate which surface to find: side or full. Therefore, we find the complete one, which includes the lateral one.
S = π * r * l + π * r² = 3.14 * 3√3 * 6 + 3.14 * (3√3) ² ≈ 3.14 * 3 * 1.73 * 6 + 3.14 * 27 ≈ 97.78 + 84.78 ≈ 182.56 (cm²).
Answer: the total surface area is 182.56 cm².