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Find the lateral area of a regular prism with height [tex]$h[tex]$[/tex] if the base of the prism is an equilateral triangle with side [tex]$[/tex]4 \, \text{cm}[tex]$[/tex] and [tex]h = 2.5 \, \text{cm}$[/tex][/tex].

Sagot :

GinnyAnswer
Let's solve this problem step by step.

### Step 1: Understanding the Problem
We are given:
- The side length of the equilateral triangle base is 4 cm.
- The height of the prism (the distance between the two triangular bases) is 2.5 cm.

We need to find the lateral area of the prism. The lateral area is the area of all the rectangular faces (sides) of the prism.

### Step 2: Calculate the Perimeter of the Equilateral Triangle
The perimeter of an equilateral triangle is found by multiplying the side length by 3.

[tex]\[ \text{Perimeter} = 3 \times 4 \text{ cm} = 12 \text{ cm} \][/tex]

### Step 3: Calculate the Lateral Area
The lateral area of the prism can be calculated by multiplying the perimeter of the base by the height of the prism.

[tex]\[ \text{Lateral Area} = \text{Perimeter of the Base} \times \text{Height of the Prism} \][/tex]

Substituting the given values:

[tex]\[ \text{Lateral Area} = 12 \text{ cm} \times 2.5 \text{ cm} = 30 \text{ cm}^2 \][/tex]

### Step 4: Result
The lateral area of the regular prism is [tex]\(\boxed{30 \text{ cm}^2}\)[/tex].
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