Volume of a Prism: Practice Questions & Formula

This page provides practice questions for calculating the volume of a prism, along with the formulas and step-by-step solutions you need to check your work. You will also find a calculator that lets you enter the base area and height to get the volume instantly.

Formula

The volume of any prism is found by multiplying the area of its base by its perpendicular height. The formula is:

V = A × h

• V = volume (cubic units, e.g., cm³, m³, in³)
• A = area of the base (square units)
• h = height of the prism, measured perpendicular to the base (linear units)

This works for all prisms—rectangular, triangular, trapezoidal, hexagonal, or any other shape—as long as the two bases are congruent parallel polygons.

Worked Example

Using the default values from the calculator above: a prism with a base area of 50 cm² and a height of 10 cm.

1. Identify the base area: A = 50 cm²
2. Identify the height: h = 10 cm
3. Apply the formula: V = A × h = 50 × 10 = 500 cm³

The volume is 500 cm³.

Practice Questions (with Answers)

Try these on your own, then check the solutions below.

1. A rectangular prism has a base length of 8 cm and a base width of 5 cm. Its height is 12 cm. What is the volume?
2. A triangular prism has a triangular base with a base of 6 m and a height of 4 m. The prism’s height is 10 m. Find the volume.
3. A trapezoidal prism has a base that is a trapezoid with parallel sides of 5 cm and 7 cm, and a distance between them of 4 cm. The prism height is 15 cm. Calculate the volume.
4. A hexagonal prism has a regular hexagon base with side length 2 cm and apothem 1.732 cm. The prism height is 8 cm. What is the volume?
5. A prism has a volume of 240 cm³ and a base area of 60 cm². What is its height?

Solutions

1. Base area = 8 × 5 = 40 cm². Volume = 40 × 12 = 480 cm³.
2. Base area = (6 × 4) / 2 = 12 m². Volume = 12 × 10 = 120 m³.
3. Trapezoid area = ((5 + 7) / 2) × 4 = 24 cm². Volume = 24 × 15 = 360 cm³.
4. Hexagon area = (perimeter × apothem) / 2 = (12 × 1.732) / 2 = 10.392 cm². Volume = 10.392 × 8 = 83.14 cm³.
5. Height = Volume / Base area = 240 / 60 = 4 cm.

How to Choose the Correct Base

The base is the two congruent parallel faces. In a rectangular prism, any face can be the base, but in a triangular or trapezoidal prism, the base is usually the triangular or trapezoidal face, not the rectangular side. Always ensure the height you use is perpendicular to that chosen base.

Common Mistakes and Misconceptions

• Using the slant height: For oblique prisms, the height must be the perpendicular distance between the bases, not the slant edge length.
• Confusing base area with a side length: The formula uses the area of the base, not its perimeter or a single dimension.
• Unit errors: If the base is in cm² and the height is in m, convert one to the other before multiplying. The volume will be in the cubed unit of the converted measurement.
• Forgetting to halve for triangles: The area of a triangular base is (base × height) / 2, a common source of errors.

Real-World Applications

Prism volume calculations appear in construction (concrete for a triangular footing), packaging (volume of a cereal box), and engineering (flow through a trapezoidal channel). Knowing how to compute volume is essential for estimating material quantities, storage capacity, or displacement.

Frequently Asked Questions

How do I find the volume of a rectangular prism?

Multiply length × width × height. Since the base is a rectangle, its area is length × width, then multiply by the height.

What is the difference between a prism and a pyramid?

A prism has two congruent parallel bases and rectangular lateral faces. A pyramid has one base and triangular lateral faces meeting at a point. Their volume formulas differ: prism is base area × height, pyramid is (1/3) × base area × height.

Can I use the same formula for a cylinder?

Yes, a cylinder is like a prism with a circular base. The volume is πr²h, which is the base area (πr²) times the height.

What if the prism is oblique (leaning)?

The formula still works, but you must use the perpendicular height between the bases, not the slant height. In a right prism, the perpendicular height equals the edge length.

How do I find the base area if I only know the volume and height?

Rearrange the formula: base area = volume / height. For example, if volume is 120 cm³ and height is 10 cm, base area = 12 cm².

Limits of the Formula

The formula V = A × h applies only to prisms (and cylinders). It does not apply to spheres, cones, or irregular shapes. Also, the base must be a flat polygon; if the cross-section changes shape along the height, it is not a prism and the formula fails.

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in a | Base area | cm² | 50 | 0
in h | Height | cm | 10 | 0
out vol | Volume | cm³ | a * h | 2