How Do You Find the Volume of the Same Prism?

To find the volume of a prism, you multiply the area of its base by its height.

The fundamental method for calculating the volume ($V$) of any prism is given by the formula:

**V = Bh**

Where:

*   **V** represents the **volume** of the prism.
*   **B** represents the **area of one of its bases**.
*   **h** represents the **height** of the prism (the perpendicular distance between the two bases).

This formula is a core concept in geometry and applies to all types of prisms, whether they have triangular, square, pentagonal, or any other polygonal base.

## Understanding the Components

*   **Base (B):** The base of a prism is one of the two congruent and parallel faces. The shape of the base determines the name of the prism (e.g., a prism with a triangular base is a triangular prism). You must calculate the area of this 2D shape first.
*   **Height (h):** The height is the perpendicular distance between the two bases. In a *right prism*, the height is the length of the edges connecting the corresponding vertices of the two bases. In an *oblique prism* (one that leans), the height is the perpendicular distance measured externally or internally, not the length of the slanting edge.

## Applying the Formula: Step-by-Step

Finding the volume of a prism involves two main steps:

1.  **Calculate the Area of the Base (B):** The method for finding 'B' depends entirely on the shape of the base.
    *   If the base is a rectangle: Area = length × width
    *   If the base is a triangle: Area = ½ × base × height (of the triangle)
    *   If the base is a circle (for a cylinder, which is a type of circular prism): Area = π × radius²
    *   For other polygons, use the specific area formula for that shape.
2.  **Multiply by the Height (h):** Once you have the area of the base (B), multiply it by the height of the prism (h).

**Example:**

Let's say you have a rectangular prism with a base that is 5 cm long and 3 cm wide, and the height of the prism is 10 cm.

1.  Calculate the area of the base (B): B = length × width = 5 cm × 3 cm = 15 cm².
2.  Calculate the volume (V): V = B × h = 15 cm² × 10 cm = 150 cm³.

The volume is 150 cubic centimeters.

## The Principle Behind the Formula

As referenced, the concept that "If two solids have the same height h and the same cross-sectional area B at every level, then they have the same volume" is based on **Cavalieri's Principle**. This principle explains why the V = Bh formula works for *any* prism (right or oblique) with the same base area and height. Imagine stacking infinitely thin slices of the base shape up to the height 'h'. The total volume is the sum of the volumes of these slices, which essentially amounts to the base area multiplied by the height.

## Summary Table

| Component | Description                                       | How to Find                                      |
| :-------- | :------------------------------------------------ | :----------------------------------------------- |
| **V**     | Volume of the Prism                               | Result of B × h                                  |
| **B**     | Area of the Prism's Base                          | Use specific area formula based on base shape    |
| **h**     | Perpendicular distance between the two bases      | Measure the height                             |

Knowing the shape of the base and the height are crucial for accurately calculating the volume of any prism using the simple yet powerful formula **V = Bh**.