The base of a right pyramid is an equilateral triangle of side `10sqrt(3) cm`. If the total surface area of the pyramid is `270 sqrt(3)`. Sq. cm, its height is

Options

A. `12sqrt(3) cm`
B. 10 cm
C. `10sqrt(3) cm`
D. 12 cm

Answer

To find the height of the right pyramid with an equilateral triangular base, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Data:**
- Side of the equilateral triangle (base) = \(10\sqrt{3}\) cm
- Total Surface Area (TSA) = \(270\sqrt{3}\) cm²

2. **Calculate the Inradius (r) of the Base Triangle:**
- The formula for the inradius \(r\) of an equilateral triangle is given by:
\[ r = \frac{a}{2\sqrt{3}} \]
where \(a\) is the side length of the triangle.
- Substituting the value of \(a\):
\[ r = \frac{10\sqrt{3}}{2\sqrt{3}} = 5 \text{ cm} \]

3. **Let the Slant Height be \(l\):**
- Using the Pythagorean theorem in the right triangle formed by the height \(h\), the inradius \(r\), and the slant height \(l\):
\[ l^2 = h^2 + r^2 \]
- Substituting \(r = 5\):
\[ l^2 = h^2 + 5^2 = h^2 + 25 \]

4. **Calculate the Perimeter of the Base Triangle:**
- The perimeter \(P\) of the equilateral triangle is:
\[ P = 3 \times a = 3 \times 10\sqrt{3} = 30\sqrt{3} \text{ cm} \]

5. **Calculate the Area of the Base Triangle:**
- The area \(A\) of the equilateral triangle is given by:
\[ A = \frac{\sqrt{3}}{4} a^2 \]
- Substituting the value of \(a\):
\[ A = \frac{\sqrt{3}}{4} (10\sqrt{3})^2 = \frac{\sqrt{3}}{4} \times 300 = 75\sqrt{3} \text{ cm}^2 \]

6. **Write the Total Surface Area Formula:**
- The formula for the total surface area (TSA) of the pyramid is:
\[ \text{TSA} = \frac{1}{2} \times P \times l + A \]
- Substituting the values we have:
\[ 270\sqrt{3} = \frac{1}{2} \times 30\sqrt{3} \times l + 75\sqrt{3} \]

7. **Simplify the Equation:**
- Simplifying the left side:
\[ 270\sqrt{3} = 15\sqrt{3} \times l + 75\sqrt{3} \]
- Rearranging gives:
\[ 15\sqrt{3} \times l = 270\sqrt{3} - 75\sqrt{3} = 195\sqrt{3} \]
- Dividing both sides by \(15\sqrt{3}\):
\[ l = \frac{195\sqrt{3}}{15\sqrt{3}} = 13 \]

8. **Substituting \(l\) Back to Find \(h\):**
- Now substituting \(l = 13\) back into the equation:
\[ 13^2 = h^2 + 25 \]
- This simplifies to:
\[ 169 = h^2 + 25 \]
- Thus:
\[ h^2 = 169 - 25 = 144 \]
- Taking the square root:
\[ h = 12 \text{ cm} \]

### Final Answer:
The height of the pyramid is \(12\) cm.