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Chapter 9: Problem 272

Find (a) the volume and (b) the surface area of the cube with the given side length. 6 inches

Short Answer

Expert verified
Volume: 216 cubic inches, Surface Area: 216 square inches.

Step by step solution

01

Identify the side length of the cube

The side length of the cube, denoted by \(s\), is given as 6 inches.
02

Calculate the volume of the cube

The formula to calculate the volume \(V\) of a cube is \(V = s^3\). Plug in the side length value: \(V = 6^3 = 6 \times 6 \times 6 = 216\) cubic inches.
03

Calculate the surface area of the cube

The formula to calculate the surface area \(SA\) of a cube is \(SA = 6s^2\). Plug in the side length value: \(SA = 6 \times 6^2 = 6 \times 36 = 216\) square inches.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Volume of a Cube
Understanding the volume of a cube is essential in geometry. The volume represents the space enclosed within a three-dimensional object. For cubes, the calculation is quite straightforward. To find the volume of a cube, one must use the formula:
\[ V = s^3 \]
Here, \( s \) stands for the side length of the cube. In essence, you multiply the side length by itself three times. This is because all sides of a cube are equal in length. For example, if the side length \( s \) is 6 inches, the volume becomes:
\[ V = 6^3 = 6 \times 6 \times 6 = 216 \text{ cubic inches} \]
This formula helps in understanding how much space the cube occupies.
Surface Area of a Cube
Next in line is the surface area, which measures the total area of all the faces of the cube. To calculate the surface area, you can use the formula:
\[ SA = 6s^2 \]
Here, \( 6 \) represents the six faces of the cube, and \( s^2 \) represents the area of one face. For a cube with side length \( s = 6 \) inches, the surface area is calculated as:
\[ SA = 6 \times 6^2 = 6 \times 36 = 216 \text{ square inches} \]
This formula demonstrates how all faces of a cube contribute to the total surface area. Knowing how to calculate surface area is crucial for various practical applications, such as determining the amount of material needed to cover the cube.
Geometry Formulas
Geometry involves many different shapes and formulas. The cube, being a fundamental shape, requires specific formulas to understand its properties correctly. Some essential geometry formulas related to cubes include:
  • Volume for a cube: \( V = s^3 \)
  • Surface area for a cube: \( SA = 6s^2 \)
These formulas are part of what makes geometry both interesting and practical.
Remember to always pay attention to units of measurement. When calculating volume, the result will be in cubic units (e.g., cubic inches). For surface area, the result will be in square units (e.g., square inches). Understanding these basic geometry formulas is a stepping stone to more complex mathematical concepts and real-life applications.

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