To find the surface area of a cone, we calculate both the lateral surface area and the base area.
- Calculate the radius of the base:
The diameter of the base is given as 25 centimeters. So, the radius \(r\) is: \[ r = \frac{\text{Diameter}}{2} = \frac{25}{2} = 12.5 \text{ centimeters} \]
- Find the height of the cone:
The height \(h\) is given as the perpendicular height from the base to the top vertex, which is: \[ h = 22 \text{ centimeters} \]
- Calculate the slant height \(l\):
The hypotenuse (slant height) of the cone is given as 22 centimeters. Thus: \[ l = 22 \text{ centimeters} \]
- Calculate the lateral surface area \(A_L\) of the cone:
The formula for the lateral surface area of a cone is: \[ A_L = \pi r l \]
Substituting the values: \[ A_L = 3.14 \times 12.5 \times 22 \] \[ A_L = 3.14 \times 275 = 865.5 \text{ square centimeters} \]
- Calculate the base area \(A_B\) of the cone:
The formula for the area of the base of a cone (which is a circle) is: \[ A_B = \pi r^2 \]
Substituting the radius: \[ A_B = 3.14 \times (12.5)^2 = 3.14 \times 156.25 = 490.625 \text{ square centimeters} \]
- Total surface area \(A_T\) of the cone:
The total surface area is the sum of the lateral surface area and the base area: \[ A_T = A_L + A_B = 865.5 + 490.625 = 1356.125 \text{ square centimeters} \]
Rounding to two decimal places, the total surface area is approximately 1,356.13 square centimeters.
Among the given choices, the closest one appears to be:
1,354.125 square centimeters
So, the correct response is: 1,354.125 square centimeters
1 answer