The cone is obtained by rotating a right-angled triangle with 6 cm and 3 cm

The cone is obtained by rotating a right-angled triangle with 6 cm and 3 cm legs around the smaller leg. Find the axial area and total surface area of the cone.

The radius of the base of the cone, obtained by rotating a right-angled triangle with 6 cm and 3 cm legs around the smaller leg, is equal to the larger leg of this triangle, i.e. r = 6 cm, the height of the cone is equal to the smaller leg, i.e. h = 3 cm.The generatrix of the cone is equal to the hypotenuse, the square of which can be found as the sum of the squares of the legs:
l² = 3² + 6² = 9 + 36 = 45;
l = √45 = 3√5.
The axial section is an isosceles triangle with a base equal to the diameter of the base of the cone and sides equal to the generatrix of the cone. Its height, drawn to the base, is equal to the height of the cone.
Ssection = 0.5 * 2r * h = r * h = 6 * 3 = 18 cm².
The total surface area of ​​the cone consists of the sum of the areas of the base and the lateral surface: S total = S main + S side.
Sb = πr² = π * 6² = 36π ≈ 113.097 cm².
The lateral surface area is equal to the product of half of the base circumference by the generatrix:
Side = πrl = π * 6 * 3√5 = 18√5π ≈ 126.447 cm².
S total = 113.097 + 126.447 = 239.544 cm².