Length of inradius of an equilateral triangle is 3 cm . Find the height of this triangle .

Answer

To find the height of an equilateral triangle given its inradius, we can follow these steps:

### Step 1: Understand the relationship between inradius and side length
The inradius \( r \) of an equilateral triangle can be expressed in terms of its side length \( a \) using the formula:
\[ r = \frac{a}{2\sqrt{3}} \]

### Step 2: Substitute the given inradius
We know that the inradius \( r \) is 3 cm. Substituting this value into the formula gives:
\[ 3 = \frac{a}{2\sqrt{3}} \]

### Step 3: Solve for the side length \( a \)
To find \( a \), we can rearrange the equation:
\[ a = 3 \times 2\sqrt{3} \]
\[ a = 6\sqrt{3} \text{ cm} \]

### Step 4: Use the side length to find the height
The height \( h \) of an equilateral triangle can be calculated using the formula:
\[ h = \frac{\sqrt{3}}{2} \times a \]
Substituting the value of \( a \):
\[ h = \frac{\sqrt{3}}{2} \times 6\sqrt{3} \]

### Step 5: Simplify the expression
Calculating the height:
\[ h = \frac{6 \times 3}{2} = \frac{18}{2} = 9 \text{ cm} \]

### Final Answer
The height of the equilateral triangle is **9 cm**.

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