If the circumradius of an equilateral triangle ABC be 8 cm, then the height of the triangle is

Options

A. 16 cm
B. 6 cm
C. 8 cm
D. 12 cm

Answer

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

### Step 1: Understand the relationship between the circumradius and the side length of the triangle.
For an equilateral triangle, the circumradius \( R \) is related to the side length \( a \) by the formula:
\[ R = \frac{a}{\sqrt{3}} \]

### Step 2: Substitute the given circumradius into the formula.
We know the circumradius \( R \) is 8 cm. Therefore, we can set up the equation:
\[ 8 = \frac{a}{\sqrt{3}} \]

### Step 3: Solve for the side length \( a \).
To find \( a \), we can rearrange the equation:
\[ a = 8 \sqrt{3} \]

### Step 4: Use the side length to find the height of the triangle.
The height \( h \) of an equilateral triangle can be calculated using the formula:
\[ h = \frac{\sqrt{3}}{2} a \]
Substituting \( a = 8 \sqrt{3} \) into the height formula:
\[ h = \frac{\sqrt{3}}{2} \times 8 \sqrt{3} \]

### Step 5: Simplify the expression for height.
Calculating the height:
\[ h = \frac{\sqrt{3} \times 8 \sqrt{3}}{2} = \frac{8 \times 3}{2} = \frac{24}{2} = 12 \text{ cm} \]

### Final Answer:
The height of the triangle is \( 12 \text{ cm} \).

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