Volume of 3D Shapes Worksheets | Questions and Revision | MME - Free Printable

Problem Analysis:

The image depicts a three-dimensional geometric shape, specifically a truncated pyramid (or frustum of a pyramid). The task is to calculate a specific property of this shape, such as its volume or surface area. Since the problem is not explicitly stated, I will assume the task is to calculate the volume of the truncated pyramid.

Known Information:

1. The dimensions of the top base are given as a square with side length \( 45 \, \text{cm} \).
2. The dimensions of the bottom base are given as a square with side length \( 60 \, \text{cm} \).
3. The height of the truncated pyramid is given as \( 20 \, \text{cm} \).
4. The slant height (the diagonal distance along the lateral face) is given as \( 80 \, \text{cm} \).

Formula for the Volume of a Truncated Pyramid:

The volume \( V \) of a truncated pyramid with square bases is given by:
\[
V = \frac{h}{3} \left( A_1 + A_2 + \sqrt{A_1 A_2} \right)
\]
where:
- \( h \) is the height of the truncated pyramid,
- \( A_1 \) is the area of the top base,
- \( A_2 \) is the area of the bottom base.

Step-by-Step Solution:

#### 1. Calculate the Area of the Top Base (\( A_1 \)):
The top base is a square with side length \( 45 \, \text{cm} \):
\[
A_1 = 45^2 = 2025 \, \text{cm}^2
\]

#### 2. Calculate the Area of the Bottom Base (\( A_2 \)):
The bottom base is a square with side length \( 60 \, \text{cm} \):
\[
A_2 = 60^2 = 3600 \, \text{cm}^2
\]

#### 3. Substitute the Values into the Volume Formula:
The height \( h \) is given as \( 20 \, \text{cm} \). Now, substitute \( h \), \( A_1 \), and \( A_2 \) into the volume formula:
\[
V = \frac{20}{3} \left( 2025 + 3600 + \sqrt{2025 \cdot 3600} \right)
\]

#### 4. Simplify the Expression Inside the Parentheses:
First, calculate the sum of the areas:
\[
2025 + 3600 = 5625
\]

Next, calculate the square root of the product of the areas:
\[
\sqrt{2025 \cdot 3600} = \sqrt{7290000} = 2700
\]

Now, add these values together:
\[
2025 + 3600 + 2700 = 8325
\]

#### 5. Calculate the Volume:
Substitute back into the volume formula:
\[
V = \frac{20}{3} \cdot 8325 = \frac{166500}{3} = 55500 \, \text{cm}^3
\]

Final Answer:

\[
\boxed{55500}
\]