Answer:
74π cm²
Explanation:
You want to know the painted area of a cone 10 cm in diameter and 12 cm high that has an unpainted hole in the base that has a 4 cm radius.
Lateral area
The area of the curved surface of the cone is found using the given formula. To use that formula, we need to know the slant height of the cone. The slant height is the hypotenuse of a right triangle that is 12 cm high and has a base of 5 cm:
l² = r² +h²
l = √(5² +12²) = √(25 +144) = √169
l = 13 . . . . cm
Then the lateral area is ...
LA = πrl = π(5 cm)(13 cm) = 65π cm²
Base area
The painted area of the base is the difference between the areas of circles 5 cm in radius and 4 cm in radius:
A = π(r₁² - r₂²) = π((5 cm)² -(4 cm)²) = π(25 -16) cm² = 9π cm²
Total painted area
The painted area is the sum of the areas of the curved surface and the donut base:
A = 65π cm² +9π cm² = 74π cm²
__
Additional comment
The radius of the base is half the diameter. If you slice the cone through its vertex and perpendicular to its base, the cross section is an isosceles triangle with height 12 cm and base 10 cm. The altitude of that triangle divides it into two congruent right triangles that have base 5 cm and height 12 cm.
You may recognize this as the {5, 12, 13} right triangle, which tells you the slant height is 13 cm. If not, you can use the Pythagorean theorem to find the slant height, as above.