Calculating Volume and Surface Area of a Modified Rectangular Prism: A Comprehensive Guide

When dealing with complex geometric shapes, understanding how to calculate their volume and surface area can be crucial. In this article, we will explore the process of determining the volume and surface area of a modified rectangular prism. Specifically, we will analyze the scenario where a cube with a side length of 2 units is removed from one of the corners of a rectangular prism with dimensions 4 x 3 x 5 units.

Step 1: Calculating the Volume of the New Object

The volume of a geometric shape is the measure of the space it occupies. We will calculate the volume of the initial rectangular prism and the removed cube, then subtract the latter from the former to determine the volume of the new object.

Step 1.1: Volume of the Rectangular Prism
The volume V_{prism} of a rectangular prism is calculated by multiplying its length, width, and height:

V_{prism} length × width × height 4 × 3 × 5 60 cubic units

Step 1.2: Volume of the Cube
The volume V_{cube} of a cube with side length 2 units is calculated by raising the side length to the third power:

V_{cube} side3 23 8 cubic units

Step 1.3: Volume of the New Object
Subtracting the volume of the cube from the volume of the rectangular prism results in the volume of the new object:

V_{new} V_{prism} - V_{cube} 60 - 8 52 cubic units

Step 2: Calculating the Surface Area of the New Object

The surface area of a geometric shape is the total area of all its faces. We will calculate the surface area of the initial rectangular prism, then adjust for the changes caused by removing the cube to find the surface area of the new object.

Step 2.1: Surface Area of the Rectangular Prism
The surface area S_{prism} of a rectangular prism is calculated using the formula:

S_{prism} 2lw 2lh 2wh

Substituting the dimensions of the prism (l 4, w 3, h 5) into the formula, we get:

S_{prism} 2(4 × 3) 2(4 × 5) 2(3 × 5) 24 40 30 94 square units

Step 2.2: Surface Area of the Cube
The surface area S_{cube} of a cube is given by:

S_{cube} 6 × side2 6 × 22 6 × 4 24 square units

Step 2.3: Adjustment for the Cut
When the cube is removed from the prism, we need to adjust the surface area by removing the area of the three faces of the cube that were part of the prism, and adding the area of the three exposed faces of the cube:

Area of 3 faces of the cube 3 × 2 × 2 12 square units

Since the new exposed faces also cover 12 square units, the total adjustment to the surface area is zero:

S_{new} S_{prism} - 12 12 94 square units

Summary

The volume of the new object is 52 cubic units, and the surface area of the new object is 94 square units.

Conclusion

This article demonstrates the step-by-step process of calculating the volume and surface area of a modified rectangular prism, illustrating how to handle more complex geometric problems.

Additional Resources

For more information on related topics, you can explore:

Rectangular Prism Calculation: Understanding the basic calculations for volume and surface area of a rectangular prism. Cube Volume and Surface Area: Detailed steps for calculating the volume and surface area of a cube. Geometric Shapes and Their Properties: A broader overview of various geometric shapes and their properties.