Certainly! Let's break down the problem and find the appropriate areas step-by-step.
1. Height and Radius of the Cone:
- The given height (h) of the cone is 16 centimeters.
- The given radius (r) of the base of the cone is 12 centimeters.
2. Slant Height Calculation:
- First, we need to determine the slant height [tex]\( l \)[/tex] of the cone. To do this, we use the Pythagorean theorem:
[tex]\[ l = \sqrt{h^2 + r^2} \][/tex]
- Substituting the given values:
[tex]\[ l = \sqrt{16^2 + 12^2} \][/tex]
[tex]\[ l = \sqrt{256 + 144} \][/tex]
[tex]\[ l = \sqrt{400} \][/tex]
Thus, the slant height [tex]\( l \)[/tex] is 20 centimeters.
3. Lateral Area Calculation:
- The formula for the lateral (or curved) surface area [tex]\( A_L \)[/tex] of a cone is:
[tex]\[ A_L = \pi r l \][/tex]
- Substituting the values:
[tex]\[ A_L = \pi \times 12 \times 20 \][/tex]
- Therefore, the lateral area [tex]\( A_L \)[/tex] is approximately 753.98 square centimeters.
4. Base Area Calculation:
- The base area [tex]\( A_B \)[/tex] of the cone is found using the formula for the area of a circle:
[tex]\[ A_B = \pi r^2 \][/tex]
- Substituting the values:
[tex]\[ A_B = \pi \times 12^2 \][/tex]
[tex]\[ A_B = \pi \times 144 \][/tex]
- Therefore, the base area [tex]\( A_B \)[/tex] is approximately 452.39 square centimeters.
5. Total Surface Area Calculation:
- The total surface area [tex]\( A_T \)[/tex] of the cone is the sum of the lateral area and the base area:
[tex]\[ A_T = A_L + A_B \][/tex]
- Substituting the values:
[tex]\[ A_T = 753.98 + 452.39 \][/tex]
- Thus, the total surface area [tex]\( A_T \)[/tex] is approximately 1206.37 square centimeters.
Therefore:
- The lateral area of the cone is 753.98 square centimeters.
- The total surface area of the cone is 1206.37 square centimeters.
