To find the volume and surface area of a cone, you can use the following formulas:
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Volume (V) of a cone: \[ V = \frac{1}{3} \pi r^2 h \] where \( r \) is the radius and \( h \) is the height.
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Surface Area (A) of a cone: \[ A = \pi r (r + l) \] where \( l \) is the slant height of the cone. The slant height can be found using the Pythagorean theorem: \[ l = \sqrt{r^2 + h^2} \]
Given:
- Radius \( r = 5 \)
- Height \( h = 12 \)
Step 1: Calculate the volume. \[ V = \frac{1}{3} \pi (5)^2 (12) = \frac{1}{3} \pi (25) (12) = \frac{1}{3} \pi (300) = 100\pi \]
Thus, the volume of the cone is approximately: \[ V \approx 100 \times 3.14 \approx 314.16 \text{ cubic units} \]
Step 2: Calculate the slant height. \[ l = \sqrt{(5)^2 + (12)^2} = \sqrt{25 + 144} = \sqrt{169} = 13 \]
Step 3: Calculate the surface area. \[ A = \pi (5) (5 + 13) = \pi (5)(18) = 90\pi \]
Thus, the total surface area of the cone is approximately: \[ A \approx 90 \times 3.14 \approx 282.6 \text{ square units} \]
Summary:
- Volume: \( 100\pi \) cubic units (approximately \( 314.16 \) cubic units)
- Total surface area: \( 90\pi \) square units (approximately \( 282.6 \) square units)