To determine the areas of a cone given the radius and slant height, you can follow a structured step-by-step approach:
1. Area of the Base:
The base of the cone is a circle. The formula to find the area of a circle is:
[tex]\[ \text{Area}_{\text{base}} = \pi \times r^2 \][/tex]
Where [tex]\( r \)[/tex] is the radius of the base.
Substituting the given radius [tex]\( r = 5 \)[/tex] cm:
[tex]\[ \text{Area}_{\text{base}} = \pi \times (5)^2 = 25\pi \text{ cm}^2 \][/tex]
2. Lateral Area:
The lateral area of a cone can be found using the formula:
[tex]\[ \text{Lateral Area} = \pi \times r \times l \][/tex]
Where [tex]\( l \)[/tex] is the slant height of the cone.
Substituting the given radius [tex]\( r = 5 \)[/tex] cm and slant height [tex]\( l = 10 \)[/tex] cm:
[tex]\[ \text{Lateral Area} = \pi \times 5 \times 10 = 50\pi \text{ cm}^2 \][/tex]
3. Surface Area:
The total surface area of the cone is the sum of the base area and the lateral area. The formula is:
[tex]\[ \text{Surface Area} = \text{Area}_{\text{base}} + \text{Lateral Area} \][/tex]
Substituting in the previously calculated values:
[tex]\[ \text{Surface Area} = 25\pi \text{ cm}^2 + 50\pi \text{ cm}^2 = 75\pi \text{ cm}^2 \][/tex]
So, the detailed solutions are:
- The area of the base of the cone is [tex]\( \boxed{25\pi \text{ cm}^2} \)[/tex].
- The lateral area of the cone is [tex]\( \boxed{50\pi \text{ cm}^2} \)[/tex].
- The surface area of the cone is [tex]\( \boxed{75\pi \text{ cm}^2} \)[/tex].
