Volume and Dimensions of a Cuboid: A Detailed Guide for SEO Optimization

Welcome to this guide on solving geometric problems involving the dimensions of a cuboid. This article is designed to help you solve problems related to the length, breadth, and height of a cuboid when given its volume. These concepts are essential for understanding basic geometry and are often covered in school curricula and standardized tests.

Introduction to Cuboids and Their Dimensions

A cuboid is a three-dimensional geometric shape with six faces, all of which are rectangles. The three dimensions of a cuboid are:

Length Breadth Height

Understanding these dimensions and how they relate to the volume of a cuboid is crucial for various applications in mathematics and real-world scenarios.

Solving the Given Problem

Let's consider the following problem: If the length of a cuboid is twice the breadth and thrice the height, and the volume of the cuboid is 972 cm3, what is the breadth of the cuboid?

Solution:

We are given the following information:

The length is twice the breadth (L 2b) The length is three times the height (L 3h) The volume of the cuboid is 972 cm3

We need to find the breadth (b).

Let's define the variables:

b breadth in cm L length in cm h height in cm

From the given relationships, we can express the length and height in terms of the breadth:

Length: L 2b
Height: h L/3 (2b)/3

The formula for the volume of a cuboid is:

Volume L × b × h

Substitute the known values into the volume formula:

972 2b × b × (2b)/3

Simplify the equation:

972 (4b3)/3

Now solve for b3:

4b3 2916

b3 729

b 9

Therefore, the breadth of the cuboid is 6 cm.

Alternative Solution Method

Let's consider another approach to solving the same problem. We are given the following:

Let the length be n. The width/breadth is n/2. The height is n/3.

Given the volume:

n × (n/2) × (n/3) 972 n3/6 972 n3 5832 n 18 n/2 9 n/3 6

Thus, the length of the cuboid is 18 cm, the breadth is 6 cm, and the height is 6 cm.

Conclusion

In conclusion, understanding and solving geometric problems like the one we discussed is essential for various applications in mathematics and real-world scenarios. By mastering such problems, students and professionals can approach more complex problems with confidence. If you need further assistance, feel free to check out our range of resources and articles on geometric problems and their solutions.

Key Takeaways

The volume of a cuboid is given by ( V L times B times H ). Given relationships between dimensions can be used to express all dimensions in terms of a single variable. Solving for unknowns using the volume equation requires manipulation of algebraic expressions.