Volume of Prisms and Cylinders

Learning Objectives

Define volume as the measure of three-dimensional space.

State the general formula for the volume of a prism (V = Base Area × height).

Calculate the volume of a rectangular prism.

Calculate the volume of a cylinder.

Understand that volume is measured in cubic units.

Measuring 3D Space

While area measures a flat, two-dimensional surface (like the floor of a room), volume measures the amount of three-dimensional space an object occupies (like all the air inside the room). Volume tells you 'how much it can hold'.

Volume is always measured in cubic units, such as cubic centimeters (cm³), cubic meters (m³), or cubic inches (in³).

Prisms

A prism is a 3D solid object with two identical and parallel bases. The other faces are parallelograms. The shape of the base gives the prism its name.

A rectangular prism has a rectangle for its base (like a shoebox).

A triangular prism has a triangle for its base.

The formula for the volume of any prism is very simple:

Volume = Area of the Base × height

(V = B × h)

Where 'B' is the area of the flat base shape, and 'h' is the height of the prism (the distance between the two bases).

Calculating the Volume of a Rectangular Prism

A rectangular prism is the most common type. Its base is a rectangle, and the area of a rectangle is length × width.

Base Area (B) = length × width (l × w)

So, the volume formula becomes:

V = (l × w) × h

Example: Find the volume of a fish tank that is 50 cm long, 20 cm wide, and 30 cm high.

Formula: V = l × w × h

Substitute: V = 50 cm × 20 cm × 30 cm

Calculate: V = 30,000 cm³

Cylinders

A cylinder is a 3D solid object with two identical and parallel circular bases. A can of soup is a cylinder. We can use the same general idea as the prism to find its volume.

Volume = Area of the Base × height

The base of a cylinder is a circle, and we know the area of a circle is A = πr².

So, the volume formula for a cylinder is:

V = (πr²) × h

Example: Find the volume of a can of soup that has a radius of 4 cm and a height of 10 cm. (Use π ≈ 3.14).

Formula: V = πr²h

Substitute: V = 3.14 × (4 cm)² × 10 cm

Solve exponent first: V = 3.14 × 16 cm² × 10 cm

Calculate: V = 502.4 cm³

Key Terms

**Volume: The amount of three-dimensional space occupied by a substance or object. It is measured in cubic units.
**Prism: A solid geometric figure whose two end faces are similar, equal, and parallel rectilinear figures, and whose sides are parallelograms.
**Rectangular Prism: A prism that has rectangular bases. Its volume is V = length × width × height.
**Cylinder: A solid geometric figure with straight parallel sides and a circular or oval cross section.
**Base (of a 3D shape): The surface a solid object stands on, or the bottom line of a shape such as a triangle or rectangle. For a prism or cylinder, there are two identical and parallel bases.
**Cubic Units: The units used to measure volume (e.g., cm³, m³, in³).