Question
Find the surface area and volume of a triangular prism with height of 12 yd & an equal square base of 8 yds using the Area of the base and all lateral faces formula for surface and Bh/3 formula for volume
Answers
GPT-4o mini
To find the surface area and volume of a triangular prism, we can follow these steps:
### Step 1: Calculate the area of the base
The base of the triangular prism is an equilateral triangle since it is stated to have an "equal square base," which often means the triangular base is equilateral given the context.
For an equilateral triangle with side length \( s \):
\[
\text{Area} = \frac{\sqrt{3}}{4} s^2
\]
Given \( s = 8 \) yards:
\[
\text{Area} = \frac{\sqrt{3}}{4} (8)^2 = \frac{\sqrt{3}}{4} \cdot 64 = 16\sqrt{3} \text{ square yards}
\]
### Step 2: Calculate the volume
The formula for the volume \( V \) of a prism is given by:
\[
V = \text{Base Area} \times \text{Height}
\]
Here, the height \( h \) of the prism is given as \( 12 \) yards. Thus, the volume is:
\[
V = 16\sqrt{3} \cdot 12 = 192\sqrt{3} \text{ cubic yards}
\]
### Step 3: Calculate the surface area
The formula for the surface area \( S \) of a prism can be calculated using:
\[
S = 2 \times \text{Base Area} + \text{Lateral Area}
\]
The lateral area of a triangular prism can be found by adding the areas of the three rectangular lateral faces. Each rectangular face has a height equal to the height of the prism, and the width is equal to a side of the triangular base.
The three rectangular lateral faces are:
1. Face corresponding to the first side of the triangle:
\[
\text{Area} = \text{side} \times \text{height} = 8 \times 12 = 96
\]
2. Face corresponding to the second side of the triangle (same as first due to symmetry):
\[
\text{Area} = 8 \times 12 = 96
\]
3. Face corresponding to the third side of the triangle (same as first and second):
\[
\text{Area} = 8 \times 12 = 96
\]
So the total lateral area is:
\[
\text{Lateral Area} = 96 + 96 + 96 = 288 \text{ square yards}
\]
### Step 4: Combine to find total surface area
The total surface area is:
\[
S = 2 \times \text{Base Area} + \text{Lateral Area} = 2 \times 16\sqrt{3} + 288
\]
Calculating \(2 \times 16\sqrt{3}\):
\[
= 32\sqrt{3}
\]
Thus,
\[
S = 32\sqrt{3} + 288 \text{ square yards}
\]
### Summary
- **Volume**: \( 192\sqrt{3} \text{ cubic yards} \)
- **Surface Area**: \( 32\sqrt{3} + 288 \text{ square yards} \)
### Step 1: Calculate the area of the base
The base of the triangular prism is an equilateral triangle since it is stated to have an "equal square base," which often means the triangular base is equilateral given the context.
For an equilateral triangle with side length \( s \):
\[
\text{Area} = \frac{\sqrt{3}}{4} s^2
\]
Given \( s = 8 \) yards:
\[
\text{Area} = \frac{\sqrt{3}}{4} (8)^2 = \frac{\sqrt{3}}{4} \cdot 64 = 16\sqrt{3} \text{ square yards}
\]
### Step 2: Calculate the volume
The formula for the volume \( V \) of a prism is given by:
\[
V = \text{Base Area} \times \text{Height}
\]
Here, the height \( h \) of the prism is given as \( 12 \) yards. Thus, the volume is:
\[
V = 16\sqrt{3} \cdot 12 = 192\sqrt{3} \text{ cubic yards}
\]
### Step 3: Calculate the surface area
The formula for the surface area \( S \) of a prism can be calculated using:
\[
S = 2 \times \text{Base Area} + \text{Lateral Area}
\]
The lateral area of a triangular prism can be found by adding the areas of the three rectangular lateral faces. Each rectangular face has a height equal to the height of the prism, and the width is equal to a side of the triangular base.
The three rectangular lateral faces are:
1. Face corresponding to the first side of the triangle:
\[
\text{Area} = \text{side} \times \text{height} = 8 \times 12 = 96
\]
2. Face corresponding to the second side of the triangle (same as first due to symmetry):
\[
\text{Area} = 8 \times 12 = 96
\]
3. Face corresponding to the third side of the triangle (same as first and second):
\[
\text{Area} = 8 \times 12 = 96
\]
So the total lateral area is:
\[
\text{Lateral Area} = 96 + 96 + 96 = 288 \text{ square yards}
\]
### Step 4: Combine to find total surface area
The total surface area is:
\[
S = 2 \times \text{Base Area} + \text{Lateral Area} = 2 \times 16\sqrt{3} + 288
\]
Calculating \(2 \times 16\sqrt{3}\):
\[
= 32\sqrt{3}
\]
Thus,
\[
S = 32\sqrt{3} + 288 \text{ square yards}
\]
### Summary
- **Volume**: \( 192\sqrt{3} \text{ cubic yards} \)
- **Surface Area**: \( 32\sqrt{3} + 288 \text{ square yards} \)