If inradius of an equilateral triangle is 3 cm. Find the height of this `Delta `

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
For an equilateral triangle, the inradius (r) is related to the side length (a) by the formula:
\[ r = \frac{a}{2\sqrt{3}} \]

### Step 2: Substitute the given inradius
We know that the inradius \( r = 3 \) cm. We can set up the equation:
\[ 3 = \frac{a}{2\sqrt{3}} \]

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

### 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
Now we simplify:
\[ h = \frac{\sqrt{3} \times 6\sqrt{3}}{2} \]
\[ h = \frac{6 \times 3}{2} \]
\[ h = \frac{18}{2} \]
\[ h = 9 \text{ cm} \]

### Final Answer
The height of the equilateral triangle is \( 9 \) cm.

---