Answer:
The lateral surface area of a cone is given by the formula:
Lateral Surface Area = π * r * L,
where π is pi (approximately 3.14159), r is the radius of the base, and L is the slant height of the cone.
The base area of a cone is given by the formula:
Base Area = π * r^2.
Given that the height (h) is 12 and the base radius (r) is 3, we can calculate the slant height (L) using the Pythagorean theorem. The slant height is the hypotenuse of a right triangle formed by the height, radius, and slant height.
Using the Pythagorean theorem:
L^2 = r^2 + h^2,
L^2 = 3^2 + 12^2,
L^2 = 9 + 144,
L^2 = 153,
L ≈ √153.
Now we can calculate the surface area of the cone:
Lateral Surface Area = π * r * L,
Lateral Surface Area = π * 3 * √153.
Base Area = π * r^2,
Base Area = π * 3^2.
To find the total surface area, we add the lateral surface area and the base area:
Surface Area = Lateral Surface Area + Base Area,
Surface Area = π * 3 * √153 + π * 3^2.
Simplifying further:
Surface Area = 3π√153 + 9π.
The surface area of the cone with a height of 12 and a base radius of 3 is approximately 3π√153 + 9π.