Question

Find the surface area and volume of a triangular prism with height of 12 yd & an equal square base of 8 yds

Answer 1

To find the surface area and volume of a triangular prism with a height of 12 yards and an equilateral triangular base where each side is 8 yards, we can follow these steps:

### Step 1: Find the area of the triangular base

An equilateral triangle's area can be calculated using the formula:

\[
A = \frac{\sqrt{3}}{4} s^2
\]

where \( s \) is the length of a side. For our triangle:

\[
s = 8 \text{ yd}
\]

Substituting the value of \( s \):

\[
A = \frac{\sqrt{3}}{4} (8)^2 = \frac{\sqrt{3}}{4} \times 64 = 16\sqrt{3} \text{ yd}^2
\]

### Step 2: Calculate the volume of the prism

The volume \( V \) of the prism can be calculated using the formula:

\[
V = \text{Base Area} \times \text{Height}
\]

Given the height of the prism is 12 yards:

\[
V = 16\sqrt{3} \times 12
\]

Calculating the volume:

\[
V = 192\sqrt{3} \text{ yd}^3
\]

### Step 3: Calculate the lateral surface area

The lateral surface area (LSA) of a prism can be determined using the formula:

\[
\text{LSA} = \text{Perimeter of Base} \times \text{Height}
\]

First, we need to find the perimeter of the triangular base. Since it is an equilateral triangle:

\[
\text{Perimeter} = 3s = 3 \times 8 = 24 \text{ yd}
\]

Substituting into the LSA formula:

\[
\text{LSA} = 24 \times 12 = 288 \text{ yd}^2
\]

### Step 4: Calculate the total surface area

The total surface area (TSA) is given by:

\[
\text{TSA} = \text{Lateral Surface Area} + 2 \times \text{Base Area}
\]

Calculating the total surface area:

\[
\text{TSA} = 288 + 2 \times 16\sqrt{3}
\]

Calculating:

\[
\text{TSA} = 288 + 32\sqrt{3} \text{ yd}^2
\]

### Summary of Results

- **Volume of the prism**: \( 192\sqrt{3} \text{ yd}^3 \)
- **Surface Area of the prism**: \( 288 + 32\sqrt{3} \text{ yd}^2 \)

These give you the final values for volume and surface area of the triangular prism.