Understanding Volume Calculations of 3D Shapes: Cubes vs. Cylinders

Learning how to calculate volume is important when we study 3D shapes, like cubes and cylinders. Each of these shapes has special features that help us figure out their volume.

Volume of a Cube

A cube is a 3D shape that has six equal square faces. Here are some key facts about a cube:

• All Sides Are Equal: All three dimensions (length, width, and height) are the same. We call this length $s$.
• Square Faces: Each face of the cube is a square.

Volume Formula

To find out the volume $V$ of a cube, we use this formula:

$$V = s^3$$

Here:

• $s$ = length of one side of the cube.

Example

Imagine a cube with a side length of 4 cm. We can find the volume like this:

$$V = 4^3 = 64 \text{ cm}^3$$

This means that if we change the side length, the volume can change a lot because of how cubes work!

Volume of a Cylinder

A cylinder is another 3D shape, but it looks different. It has two circular ends and a curved side connecting them. Here are the main features of a cylinder:

• Circular Bases: It has two circles that are the same size and are parallel to each other.
• Height: This is the distance between the two circular ends, which we call $h$.
• Radius: The radius is the distance from the center to the edge of one of the circular bases, which we call $r$.

Volume Formula

To calculate the volume $V$ of a cylinder, we use this formula:

$$V = \pi r^2 h$$

Here:

• $r$ = radius of the base,
• $h$ = height of the cylinder,
• $\pi \approx 3.14$ (a number we often use in math).

Example

For a cylinder that has a radius of 3 cm and a height of 5 cm, we can find the volume like this:

$$V = \pi (3^2)(5) = \pi (9)(5) \approx 3.14 \times 45 \approx 141.3 \text{ cm}^3$$

Key Differences in Volume Calculation

1. Shape and Size:

   • A cube's volume only depends on one measurement (the side length).
   • A cylinder's volume depends on both the radius and the height.

2. Growing Volume:

   • The volume of a cube increases a lot when you make the side longer.
   • For a cylinder, when you grow the radius, the base area increases a lot, and making the height taller doubles the volume.

3. Changes in Dimensions:

   • If you double the side length of a cube ($s \rightarrow 2s$), the volume becomes eight times bigger ($V \rightarrow 8V$).
   • For a cylinder, if you double the radius, the base area gets four times bigger, and if you double the height, the volume doubles. So if you double both, the volume also becomes eight times bigger.

4. Calculating Volume:

   • Finding the volume of a cube is simple and only takes one step using the side length.
   • The volume of a cylinder is more complex because we need to square the radius and then multiply it by the height.

In summary, cubes and cylinders are both important 3D shapes, but they require different ways to calculate volume. Understanding these differences is important for students learning math in Year 7.