The base of a solid right prism is a triangle whose sides are 9 cm, 12 cm and 15 cm, The height of the prism is 5 cm. Then the total surface area of the prism is

Options

A. 180 sq cm
B. 234 sq cm
C. 270 sq cm
D. 288 sq cm

Answer

To find the total surface area of the solid right prism with a triangular base, we will follow these steps:

### Step 1: Identify the sides of the triangle
The sides of the triangle are given as 9 cm, 12 cm, and 15 cm. We can confirm that this is a right triangle since these sides satisfy the Pythagorean theorem (9² + 12² = 15²).

### Step 2: Calculate the area of the triangular base
For a right triangle, we can use the formula for the area:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
Here, we can take the base as 12 cm and the height as 9 cm. Thus,
\[ \text{Area} = \frac{1}{2} \times 12 \times 9 = \frac{108}{2} = 54 \text{ cm}^2 \]

### Step 3: Calculate the perimeter of the triangular base
The perimeter of the triangle can be calculated by adding all the sides together:
\[ \text{Perimeter} = 9 + 12 + 15 = 36 \text{ cm} \]

### Step 4: Use the formula for the total surface area of the prism
The total surface area (TSA) of a prism is given by the formula:
\[ \text{TSA} = \text{Perimeter of base} \times \text{Height} + 2 \times \text{Area of base} \]
Substituting the values we have:
\[ \text{TSA} = 36 \times 5 + 2 \times 54 \]

### Step 5: Calculate the total surface area
Calculating each part:
1. \(36 \times 5 = 180\)
2. \(2 \times 54 = 108\)

Now, adding these two results together:
\[ \text{TSA} = 180 + 108 = 288 \text{ cm}^2 \]

### Final Result
The total surface area of the prism is \(288 \text{ cm}^2\).

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