Welcome back to Volume of 3D Solids! In the previous lesson, you learned how to calculate the volume of rectangular prisms and cubes using the formula $V = \text{length} \times \text{width} \times \text{height}$. This lesson focuses on understanding volume as base area times height. You will now use what you know about this familiar three-number multiplication to understand why it is really a two-step idea: find the area of the base, then multiply by the height.

By the end of this lesson, you will have learned how to:

• Rewrite the rectangular prism volume formula as $V = B \times h$.
• Explain why the volume of a uniform solid equals the area of its base times its height using the concept of stacked layers.
• Calculate the volume of a solid given its base area and height, expressing the answer in correct cubic units.
• Describe how changing the base area or the height affects the total volume proportionally.

Before you touch any formulas, let's revisit a mental picture from the last lesson. Recall how you imagined filling a box with unit cubes: first, laying a single flat layer across the bottom, and then stacking identical layers until the box was full.

That bottom layer is the key. Its size is determined by the shape and area of the base of the solid, and the number of layers equals the height. Think of a stack of identical playing cards where each card represents one thin layer, and the height of the stack tells you how many cards there are. If you keep this "layers" image in mind, the rest of this lesson will feel very natural.

The volume of a rectangular prism is:

Now let's group the first two factors together. The product $\text{length} \times \text{width}$ gives you the area of the rectangular base. You'll call that quantity $$ (for base area). The formula then becomes:

Understanding the formula $V = B \times h$ also helps you predict what happens when dimensions change. Because volume is the product of two factors, changing either one has a direct and proportional effect.

Changing the height while keeping the base area the same: Imagine two rectangular prisms that share the same 3 ft × 4 ft base ($B = 12 \ \text{ft}^2$). One is 2 ft tall and the other is 5 ft tall. The shorter prism has , while the taller prism has . The taller prism holds more than double the volume because it contains more layers of the same base area. In general, if you double the height, you double the volume; triple the height, triple the volume.

Here is a compact summary of the key ideas from this lesson:

The volume formula $V = \text{length} \times \text{width} \times \text{height}$ is really a special case of the more general formula $V = B \times h$, where is the area of the base. This works because a prism's volume is built from identical layers stacked to its full height. Changing either the base area or the height changes the volume proportionally — double one factor while holding the other constant, and the total volume doubles.