Question

What is the volume of this cube if each side is 3 ft long?

Short answer

The volume of the cube is 27 cubic feet.

Step-by-step solution

Identify Cube Side Length

The problem states that each side of the cube is 3 feet long.

Recall Volume Formula for Cube

The formula for the volume of a cube is \[ V = s^3 \] where \( s \) is the length of a side of the cube.

Substitute the Side Length

Substitute the given side length (3 feet) into the formula: \[ V = 3^3 \]

Calculate the Volume

Calculate the volume: \[ V = 3 \times 3 \times 3 = 27 \text{ cubic feet} \]

Key concepts

Geometry

Geometry is a branch of mathematics that deals with shapes, sizes, and the properties of space. One important concept in geometry is understanding different types of shapes and their characteristics.

Cubes are three-dimensional shapes that have equal-length sides. In the case of our exercise, we are working with a cube, which is a special type of rectangular prism where all sides are of equal length.

When working on problems involving geometric shapes, it's essential first to identify the type of shape you're dealing with. Knowing that a shape is a cube gives you important clues about its properties, such as all its faces being squares and its volume being calculated in a specific way.

Volume Formula

Calculating the volume of a shape helps us understand how much space it occupies. For a cube, the volume formula is \(V = s^3\), where \(s\) represents the length of each side of the cube.

This formula works because a cube's volume is essentially found by multiplying its length by its width and its height. Since all sides of a cube are the same, this calculation simplifies to raising one side length to the power of three.

To provide a practical example: In the exercise, each side of the cube is given as 3 feet. By substituting this value into the volume formula \[V = 3^3 = 27 \text{ cubic feet}\]. It shows that the cube's volume is 27 cubic feet, meaning it could hold 27 one-cubic-foot boxes inside.

Mathematics Education

Mathematics education is more than just learning formulas; it's about understanding concepts and applying them to solve problems. When tackling exercise problems like calculating the volume of a cube, it's crucial to follow a logical series of steps.

Start by identifying what you're given in the problem and what you need to find. This approach will help you set up your equations correctly.

By grasping the volume formula and understanding geometric properties, you're not just memorizing steps but learning to think critically and systematically. This forms a foundation for tackling more complex mathematical tasks in the future.

Encourage yourself to practice regularly. Applying these concepts in different scenarios will solidify your understanding and build confidence in your mathematical abilities.