The radius of the circle circumscribing the three vertices of an equilateral triangle is 12 cm. Determine the height (in cm.) of the equilateral triangle.

Options

A. 12
B. 14
C. 18
D. 20

Answer

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

### Step 1: Understand the relationship between the circumradius and the side length of the equilateral triangle.

For an equilateral triangle, the relationship between the circumradius \( R \) and the side length \( a \) is given by the formula:
\[ R = \frac{a}{\sqrt{3}} \]

### Step 2: Substitute the given circumradius into the formula.

Given that the circumradius \( R \) is 12 cm, we can substitute this value into the formula:
\[ 12 = \frac{a}{\sqrt{3}} \]

### Step 3: Solve for the side length \( a \).

To find \( a \), we can rearrange the equation:
\[ a = 12 \sqrt{3} \]

### Step 4: Calculate the height of the equilateral triangle.

The height \( h \) of an equilateral triangle can be found using the formula:
\[ h = \frac{\sqrt{3}}{2} a \]
Substituting \( a = 12 \sqrt{3} \) into the height formula:
\[ h = \frac{\sqrt{3}}{2} \times 12 \sqrt{3} \]

### Step 5: Simplify the expression for height.

Calculating the height:
\[ h = \frac{\sqrt{3} \times 12 \sqrt{3}}{2} = \frac{12 \times 3}{2} = \frac{36}{2} = 18 \text{ cm} \]

### Final Answer:
The height of the equilateral triangle is \( 18 \) cm.

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