https://file.aiquickdraw.com/mathbot/file/1725906027596ep1zxhy1.jpg

To calculate the volume of the frustum, we need to follow these steps:

1. **Volume of the entire pyramid**:  
   The volume \( V \) of a pyramid is given by the formula:
   \[
   V = \frac{1}{3} \times \text{Area of base} \times \text{Height}
   \]
   For the given square-based pyramid:
   - The area of the base \( A_b \) is \( 8 \, \text{cm} \times 8 \, \text{cm} = 64 \, \text{cm}^2 \).
   - The height \( h \) of the pyramid is \( 30 \, \text{cm} \).

   So, the volume of the entire pyramid \( V_{\text{pyramid}} \) is:
   \[
   V_{\text{pyramid}} = \frac{1}{3} \times 64 \, \text{cm}^2 \times 30 \, \text{cm} = 640 \, \text{cm}^3
   \]

2. **Volume of the smaller pyramid (cut-off part)**:  
   The smaller pyramid that is cut off has a height of \( 15 \, \text{cm} \). Since the height is halved, and the pyramid is similar, the side length of the smaller pyramid's base will also be halved, i.e., \( 4 \, \text{cm} \).

   The area of the base of the smaller pyramid:
   \[
   A_{\text{small base}} = 4 \, \text{cm} \times 4 \, \text{cm} = 16 \, \text{cm}^2
   \]

   The volume of the smaller pyramid \( V_{\text{small pyramid}} \) is:
   \[
   V_{\text{small pyramid}} = \frac{1}{3} \times 16 \, \text{cm}^2 \times 15 \, \text{cm} = 80 \, \text{cm}^3
   \]

3. **Volume of the frustum**:  
   The volume of the frustum is the difference between the volume of the entire pyramid and the volume of the smaller pyramid:
   \[
   V_{\text{frustum}} = V_{\text{pyramid}} - V_{\text{small pyramid}} = 640 \, \text{cm}^3 - 80 \, \text{cm}^3 = 560 \, \text{cm}^3
   \]

Thus, the volume of the frustum is **560 cm³**.

---

Would you like more details or have any questions?

Here are some related questions to explore:
1. How does the volume change if the frustum's height is doubled?
2. What would be the volume if the original pyramid had a different base area?
3. How would the formula change if the base was a rectangle instead of a square?
4. How can the surface area of the frustum be calculated?
5. What is the effect on the volume if the frustum is cut at a different height?

**Tip:** Remember that in similar shapes, corresponding linear dimensions (like heights and side lengths) scale by the same factor, but areas scale by the square of that factor, and volumes scale by the cube of that factor.

Calculate the volume of a frustum formed by cutting a square-based pyramid with a base side of 8 cm and a height of 30 cm horizontally at 15 cm.

Learn how to calculate the volume of a frustum formed by cutting a square-based pyramid with a base of 8 cm and a height of 30 cm. Understand the geometric concepts involved and apply the formula for the volume of pyramids and frustums.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation