Use the image to answer the question.
A cone shows a base diameter of 25 centimeters, perpendicular height from the base to the top vertex at 90 degrees, and 22 centimeters as the hypotenuse or the side of the cone.
What is the surface area of the cone? Use 3.14 for pi.
(1 point)
To find the surface area of the cone, we need to calculate the lateral surface area and the base area separately.
The lateral surface area can be calculated using the formula:
Lateral Surface Area = π * r * l,
where r is the radius of the base and l is the slant height of the cone.
Given that the diameter of the base is 25 centimeters, the radius is half of that, so r = 25/2 = 12.5 centimeters.
The slant height, l, can be found using the Pythagorean Theorem:
l = √(h^2 + r^2),
where h is the height of the cone.
Given that the hypotenuse or the side of the cone is 22 centimeters, we can use the Pythagorean Theorem to find the height:
22^2 = h^2 + 12.5^2,
484 = h^2 + 156.25,
h^2 = 484 - 156.25,
h^2 = 327.75,
h = √327.75,
h ≈ 18.11 centimeters.
Plugging in the values into the formula for the lateral surface area, we get:
Lateral Surface Area = π * 12.5 * 18.11 ≈ 717.58 square centimeters.
The base area of the cone can be calculated using the formula:
Base Area = π * r^2,
where r is the radius of the base.
Plugging in the value for the radius, we get:
Base Area = π * (12.5)^2 = π * 156.25 ≈ 490.87 square centimeters.
Therefore, the surface area of the cone is the sum of the lateral surface area and the base area:
Surface Area = Lateral Surface Area + Base Area
Surface Area ≈ 717.58 + 490.87
Surface Area ≈ 1208.45 square centimeters.