To find the surface area of a cone, we can use the formula:
\[ \text{Surface Area} = \pi r^2 + \pi r l \]
where:
- \( r \) is the radius of the base of the cone,
- \( l \) is the slant height of the cone.
First, we need to find the radius \( r \) and the slant height \( l \).
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Calculate the radius: \[ r = \frac{\text{diameter}}{2} = \frac{25 , \text{cm}}{2} = 12.5 , \text{cm} \]
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Calculate the slant height \( l \) using the Pythagorean theorem: \[ l = \sqrt{r^2 + h^2} \] where \( h \) is the height of the cone. Substituting the values: \[ l = \sqrt{(12.5)^2 + (22)^2} \] \[ l = \sqrt{156.25 + 484} = \sqrt{640.25} \approx 25.3 , \text{cm} \]
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Calculate the surface area: Now, plug the values into the surface area formula: \[ \text{Surface Area} = \pi r^2 + \pi r l \] \[ = 3.14 \times (12.5)^2 + 3.14 \times (12.5) \times (25.3) \] \[ = 3.14 \times 156.25 + 3.14 \times 12.5 \times 25.3 \] Calculate each term:
- \( 3.14 \times 156.25 \approx 490.625 \)
- \( 3.14 \times 12.5 \times 25.3 \approx 3.14 \times 316.25 \approx 995.375 \)
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Add the two results: \[ \text{Surface Area} \approx 490.625 + 995.375 = 1486 \]
Therefore, the surface area of the cone is approximately 1486 cm².
2 answers