https://file.aiquickdraw.com/mathbot/file/1728578057924p2uyzd8d.jpg

This problem asks you to find the area of the diagonal cross-section of a rectangular prism (right quadrangular prism). Let's break down the given information:

1. The base is a square with side length \( 3\sqrt{2} \, \text{cm} \).
2. The height (lateral edge) of the prism is \( 5 \, \text{cm} \).

The diagonal cross-section of a right quadrangular prism is a rectangle formed by the diagonal of the square base and the height of the prism. To calculate the area of this rectangle:

### Step 1: Find the diagonal of the square base
The side of the square base is \( 3\sqrt{2} \, \text{cm} \), so we use the Pythagorean theorem to find the diagonal \( d \):

\[
d = \sqrt{(3\sqrt{2})^2 + (3\sqrt{2})^2} = \sqrt{18 + 18} = \sqrt{36} = 6 \, \text{cm}
\]

### Step 2: Calculate the area of the diagonal cross-section
The diagonal cross-section is a rectangle with one side as the diagonal of the square base (6 cm) and the other side as the height of the prism (5 cm). The area \( A \) is:

\[
A = \text{diagonal} \times \text{height} = 6 \times 5 = 30 \, \text{cm}^2
\]

### Final Answer:
The area of the diagonal cross-section is \( 30 \, \text{cm}^2 \).

Would you like more details or have any questions?

Here are some related questions:
1. How do you calculate the diagonal of a rectangle?
2. What is the significance of the Pythagorean theorem in geometry?
3. How does the area of the cross-section change if the lateral edge is doubled?
4. What would happen to the cross-sectional area if the base shape were a rectangle instead of a square?
5. Can you explain how to find the volume of this prism?

**Tip**: Remember that the diagonal of a square can always be calculated using the formula \( d = s\sqrt{2} \), where \( s \) is the side length.

У правильній чотирикутній призмі сторона основи дорівнює 3√2 см, а бічне ребро — 5 см. Знайдіть площу діагонального перерізу призми.

Learn how to calculate the area of a diagonal cross-section of a quadrangular prism where the base side length is 3√2 cm and the height is 5 cm. Using the Pythagorean theorem, we find the diagonal and then calculate the cross-sectional area. Suitable for students in Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation