Question

9 in ->
6 in
8 in

Find the Volume of this triangular prism.



(4 points)
height of the triangular Base =
inches

area of the triangular Base =
square inches

height of the triangular prism =
inches

Volume of the triangular prism =
cubic inches

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Answer 1

To find the volume of a triangular prism, we need to know the area of the triangular base and the height of the prism.

### Given Dimensions:
1. The triangular base has one side measuring 9 inches, another side measuring 6 inches, and the last side measuring 8 inches.
2. The height of the triangular prism (the length of the prism) is not explicitly given in the information you provided, but let's assume it as 'h' for now.

### Step 1: Calculate the Area of the Triangular Base

To find the area of the triangular base, we can use Heron's Formula.

1. **Calculate the semi-perimeter (s)**:
\[
s = \frac{a + b + c}{2} = \frac{9 + 6 + 8}{2} = \frac{23}{2} = 11.5 \text{ inches}
\]

2. **Determine the area (A)** using Heron's Formula:
\[
A = \sqrt{s(s-a)(s-b)(s-c)}
\]
where \( a = 9 \), \( b = 6 \), and \( c = 8 \).

Now applying the values:
\[
A = \sqrt{11.5 \times (11.5 - 9) \times (11.5 - 6) \times (11.5 - 8)}
\]
\[
= \sqrt{11.5 \times 2.5 \times 5.5 \times 3.5}
\]
First let's calculate:
\[
11.5 - 9 = 2.5, \quad 11.5 - 6 = 5.5, \quad 11.5 - 8 = 3.5
\]
Now calculate the product:
\[
= 11.5 \times 2.5 = 28.75
\]
\[
= 28.75 \times 5.5 \approx 158.125
\]
\[
= 158.125 \times 3.5 \approx 553.4375
\]
The area is:
\[
A \approx \sqrt{553.4375} \approx 23.5 \, \text{square inches}
\]

### Step 2: Determine the Height of the Prism
Let’s denote the height of the prism (the length of the prism) as \( h \) (not specifically provided).

### Step 3: Calculate the Volume
The volume \( V \) of the triangular prism is given by the formula:
\[
V = A \times h
\]
Substituting \( A \) and height \( h \):
\[
V \approx 23.5 \times h \, \text{cubic inches}
\]

### Summary
- Height of the triangular base: **not specifically defined** but you will need to choose one if necessary.
- Area of the triangular base: **approximately 23.5 square inches**
- Height of the triangular prism: **h (as given or required)**
- Volume of the triangular prism: **approximately \( 23.5h \) cubic inches**

Please replace \( h \) with the actual height of the prism if it is provided.