Sure, let's break down the problem step by step:
1. Area of the Base of the Cone:
The base of the cone is a circle with radius [tex]\( r = 5 \)[/tex] cm. The area [tex]\( A \)[/tex] of a circle is calculated using the formula:
[tex]\[ A = \pi r^2 \][/tex]
Plugging in the radius:
[tex]\[ A = \pi (5)^2 = \pi \times 25 = 25 \pi \text{ cm}^2 \][/tex]
Therefore, the area of the base of the cone is [tex]\( \boxed{25} \pi \text{ cm}^2 \)[/tex].
2. Lateral Area of the Cone:
The lateral area [tex]\( L \)[/tex] of a cone is calculated using the formula:
[tex]\[ L = \pi r l \][/tex]
where [tex]\( l \)[/tex] is the slant height. Given [tex]\( r = 5 \)[/tex] cm and [tex]\( l = 10 \)[/tex] cm:
[tex]\[ L = \pi \times 5 \times 10 = 50 \pi \text{ cm}^2 \][/tex]
Therefore, the lateral area of the cone is [tex]\( \boxed{50} \pi \text{ cm}^2 \)[/tex].
3. Surface Area of the Cone:
The total surface area [tex]\( S \)[/tex] of a cone is the sum of the area of the base and the lateral area. Thus:
[tex]\[ S = \pi r^2 + \pi r l \][/tex]
We already have:
[tex]\[ \pi r^2 = 25 \pi \][/tex]
[tex]\[ \pi r l = 50 \pi \][/tex]
So, combining these:
[tex]\[ S = 25 \pi + 50 \pi = 75 \pi \text{ cm}^2 \][/tex]
Therefore, the surface area of the cone is [tex]\( \boxed{75} \pi \text{ cm}^2 \)[/tex].
