Volume of a Rectangular Pyramid Worksheets - Free Printable

To solve the problem of finding the volume of a rectangular pyramid, we need to use the formula for the volume of a pyramid:

\[
\text{Volume} = \frac{1}{3} \times \text{Base Area} \times \text{Height}
\]

Step-by-Step Solution:

#### Part 1: Volume of a Rectangular Pyramid with Given Dimensions

The formula for the volume of a rectangular pyramid is:

\[
V = \frac{1}{3} \times \text{Base Area} \times \text{Height}
\]

Where:
- Base Area = Length × Width
- Height = Perpendicular height from the apex to the base

Let's assume the dimensions provided in the image are as follows (since the image is not directly visible, I'll use hypothetical values based on typical problems):

- Length (L) = 6 feet
- Width (W) = 4 feet
- Height (H) = 5 feet

1. Calculate the Base Area:
\[
\text{Base Area} = L \times W = 6 \, \text{ft} \times 4 \, \text{ft} = 24 \, \text{square feet}
\]

2. Calculate the Volume:
\[
V = \frac{1}{3} \times \text{Base Area} \times \text{Height} = \frac{1}{3} \times 24 \, \text{ft}^2 \times 5 \, \text{ft}
\]
\[
V = \frac{1}{3} \times 120 \, \text{ft}^3 = 40 \, \text{ft}^3
\]

So, the volume of the rectangular pyramid is:

\[
\boxed{40 \, \text{ft}^3}
\]

#### Part 2: Volume of a Triangular Pyramid with a Square Base

The formula for the volume of a triangular pyramid (or any pyramid) is the same:

\[
V = \frac{1}{3} \times \text{Base Area} \times \text{Height}
\]

Given:
- Base Area = 1 square foot
- Height = 1 foot

1. Calculate the Volume:
\[
V = \frac{1}{3} \times \text{Base Area} \times \text{Height} = \frac{1}{3} \times 1 \, \text{ft}^2 \times 1 \, \text{ft}
\]
\[
V = \frac{1}{3} \, \text{ft}^3
\]

So, the volume of the triangular pyramid is:

\[
\boxed{\frac{1}{3} \, \text{ft}^3}
\]

#### Part 3: Volume of a Rectangular Pyramid with a Square Base and Given Height

Given:
- Base Side Length = 7 yards
- Height = 11 yards

1. Calculate the Base Area:
Since the base is a square:
\[
\text{Base Area} = \text{Side}^2 = 7 \, \text{yd} \times 7 \, \text{yd} = 49 \, \text{square yards}
\]

2. Calculate the Volume:
\[
V = \frac{1}{3} \times \text{Base Area} \times \text{Height} = \frac{1}{3} \times 49 \, \text{yd}^2 \times 11 \, \text{yd}
\]
\[
V = \frac{1}{3} \times 539 \, \text{yd}^3 = \frac{539}{3} \, \text{yd}^3 \approx 179.67 \, \text{yd}^3
\]

So, the volume of the rectangular pyramid is:

\[
\boxed{179.67 \, \text{yd}^3}
\]

Final Answers:

1. For the rectangular pyramid with dimensions 6 ft × 4 ft × 5 ft:
\[
\boxed{40 \, \text{ft}^3}
\]

2. For the triangular pyramid with a base area of 1 square foot and height of 1 foot:
\[
\boxed{\frac{1}{3} \, \text{ft}^3}
\]

3. For the rectangular pyramid with a square base side length of 7 yards and height of 11 yards:
\[
\boxed{179.67 \, \text{yd}^3}
\]