Question
Find the volume of the rectangular pyramid. (Triangle: 6ft, 7ft, 2ft)
Answers
GPT-4o mini
To find the volume of a rectangular pyramid, we first need to identify the base area and the height of the pyramid. The formula for the volume \( V \) of a rectangular pyramid is:
\[
V = \frac{1}{3} \times \text{Base Area} \times \text{Height}
\]
### Step 1: Identify the Base Area
You mentioned dimensions that appear to be for a triangle (6 ft, 7 ft, 2 ft). However, a rectangular pyramid has a rectangular base. Let's assume you meant the dimensions for a triangular base (with a height of 2 ft) and that the base is a triangle.
To calculate the area \( A \) of the triangular base, we can use the formula for the area of a triangle:
\[
A = \frac{1}{2} \times \text{base} \times \text{height}
\]
Here, we can identify the base of the triangle as either 6 ft or 7 ft. To keep with conventional calculations, let's take the base as 6 ft and the height as 2 ft.
### Calculation of Triangle Area
- Base = 6 ft
- Height = 2 ft
\[
A = \frac{1}{2} \times 6 \, \text{ft} \times 2 \, \text{ft} = \frac{1}{2} \times 12 \, \text{ft}^2 = 6 \, \text{ft}^2
\]
### Step 2: Use the Given Height
Since the height of the pyramid is given as 7 ft, we can now use this in our volume formula.
### Step 3: Calculate the Volume
Now we can calculate the volume:
\[
V = \frac{1}{3} \times \text{Base Area} \times \text{Height}
\]
Substituting the values we calculated:
\[
V = \frac{1}{3} \times 6 \, \text{ft}^2 \times 7 \, \text{ft} = \frac{1}{3} \times 42 \, \text{ft}^3 = 14 \, \text{ft}^3
\]
### Final Answer
The volume of the rectangular pyramid is:
\[
\boxed{14} \, \text{ft}^3
\]
\[
V = \frac{1}{3} \times \text{Base Area} \times \text{Height}
\]
### Step 1: Identify the Base Area
You mentioned dimensions that appear to be for a triangle (6 ft, 7 ft, 2 ft). However, a rectangular pyramid has a rectangular base. Let's assume you meant the dimensions for a triangular base (with a height of 2 ft) and that the base is a triangle.
To calculate the area \( A \) of the triangular base, we can use the formula for the area of a triangle:
\[
A = \frac{1}{2} \times \text{base} \times \text{height}
\]
Here, we can identify the base of the triangle as either 6 ft or 7 ft. To keep with conventional calculations, let's take the base as 6 ft and the height as 2 ft.
### Calculation of Triangle Area
- Base = 6 ft
- Height = 2 ft
\[
A = \frac{1}{2} \times 6 \, \text{ft} \times 2 \, \text{ft} = \frac{1}{2} \times 12 \, \text{ft}^2 = 6 \, \text{ft}^2
\]
### Step 2: Use the Given Height
Since the height of the pyramid is given as 7 ft, we can now use this in our volume formula.
### Step 3: Calculate the Volume
Now we can calculate the volume:
\[
V = \frac{1}{3} \times \text{Base Area} \times \text{Height}
\]
Substituting the values we calculated:
\[
V = \frac{1}{3} \times 6 \, \text{ft}^2 \times 7 \, \text{ft} = \frac{1}{3} \times 42 \, \text{ft}^3 = 14 \, \text{ft}^3
\]
### Final Answer
The volume of the rectangular pyramid is:
\[
\boxed{14} \, \text{ft}^3
\]