To calculate the surface area of a cone, we need to find both the lateral surface area and the base area. The formula for the surface area of a cone is:
\[ \text{Surface Area} = \text{Lateral Surface Area} + \text{Base Area} \]
Step 1: Calculate the radius and height
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Radius (r): The diameter is given as 25 cm, so the radius is: \[ r = \frac{\text{diameter}}{2} = \frac{25}{2} = 12.5 \text{ cm} \]
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Hypotenuse (slant height, l): The slant height is given as 22 cm.
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Height (h): We can find the height of the cone using the Pythagorean theorem. The relationship is: \[ l^2 = r^2 + h^2 \] Plugging in the values: \[ 22^2 = 12.5^2 + h^2 \] \[ 484 = 156.25 + h^2 \] \[ h^2 = 484 - 156.25 = 327.75 \] \[ h = \sqrt{327.75} \approx 18.11 \text{ cm} \]
Step 2: Calculate the lateral surface area
The formula for the lateral surface area of a cone is: \[ \text{Lateral Surface Area} = \pi r l \] Substituting the known values: \[ \text{Lateral Surface Area} = 3.14 \times 12.5 \times 22 \] Calculating: \[ \text{Lateral Surface Area} = 3.14 \times 275 = 865.5 \text{ cm}^2 \]
Step 3: Calculate the base area
The formula for the area of the base (which is a circle) is: \[ \text{Base Area} = \pi r^2 \] Substituting the known values: \[ \text{Base Area} = 3.14 \times (12.5)^2 \] Calculating: \[ \text{Base Area} = 3.14 \times 156.25 = 490.625 \text{ cm}^2 \]
Step 4: Calculate the total surface area
Now, combine the lateral surface area and the base area: \[ \text{Surface Area} = \text{Lateral Surface Area} + \text{Base Area} \] \[ \text{Surface Area} = 865.5 + 490.625 = 1356.125 \text{ cm}^2 \]
Final Result
The surface area of the cone is approximately: \[ \boxed{1356.13 \text{ cm}^2} \]
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