Grade 8 lesson: Rectangular prism volume formulas

Position in the unit

This record is the dedicated lesson for Grade 8 Geometry and Measurement: Volume Formula Fundamentals: Rectangular Prism Volume Formulas.

Use these linked records alongside this lesson rather than as substitutes for it:

• Unit overview: rea.m08.geometry-measurement.volume.basic-formulas.rectangular-prisms
• Broader lesson across prism and cylinder formulas: rea.m08.geometry-measurement.volume.basic-formulas.lesson
• Misconceptions and repairs: rea.m08.geometry-measurement.volume.basic-formulas.misconceptions
• Cumulative assessment: rea.m08.geometry-measurement.volume.basic-formulas.unit-mastery-quiz

Learning goal

By the end of this lesson, you should be able to:

• explain volume as the number of cubic units that fill a rectangular prism,
• use V = l × w × h and V = B × h correctly,
• write answers with correct cubic units,
• solve for a missing dimension when volume is known.

1. Concept development: why the formula makes sense

A rectangular prism is a box-shaped solid with rectangular faces. Its volume tells how much three-dimensional space it occupies.

Imagine filling the prism with unit cubes of side length 1 unit.

• One layer on the bottom has l × w cubes because the base is arranged in rows and columns.
• If there are h identical layers stacked vertically, the total number of cubes is (l × w) × h.

So volume is really structured counting:

• cubes in one layer, times
• number of layers.

This gives the formula V = l × w × h

A second way to say the same thing is:

• first find the area of the base,
• then multiply by the height.

If base area is B, then V = B × h

These two formulas are equivalent for a rectangular prism because, for a rectangular base, B = l × w.

2. Visual description

Picture a cereal box.

• The length is one horizontal edge of the base.
• The width is the other horizontal edge of the base.
• The height is the vertical distance from bottom to top.

If the base has 6 cubes across and 4 cubes deep, one layer contains 6 × 4 = 24 cubes. If the box is 5 cubes tall, then there are 5 such layers: 24 + 24 + 24 + 24 + 24 = 5 × 24 = 120 cubes.

That is the volume: 120 cubic units.

3. Notation and formulas

Standard notation

• V = volume
• l = length
• w = width
• h = height
• B = area of the base

Formulas

For a rectangular prism:

• V = l × w × h
• V = B × h

For a prism with a rectangular base:

• B = l × w

So:

• V = (l × w) × h
• V = lwh

4. Units

Volume must be written in cubic units.

Examples:

• centimetres -> cm^3
• metres -> m^3
• inches -> in^3

Why cubic units? Because volume counts 3D units such as 1 cm × 1 cm × 1 cm cubes.

Important distinction:

• area uses square units like cm^2,
• volume uses cubic units like cm^3.

5. Procedure: finding volume

Method A: use length, width, and height

1. Identify l, w, and h.
2. Check that all measurements use the same unit.
3. Multiply: V = l × w × h.
4. Write the answer in cubic units.

Method B: use base area times height

1. Identify the base.
2. Find the base area B.
3. Multiply by the height: V = B × h.
4. Write the answer in cubic units.

6. Procedure: finding a missing dimension

If one dimension is missing and volume is known:

1. Write the formula V = lwh.
2. Substitute the known values.
3. Divide by the product of the known dimensions.
4. Check the unit of the missing dimension is a length unit, not a cubic unit.

Example structure: If V = 180 cm^3, l = 9 cm, and w = 4 cm, then 180 = 9 × 4 × h 180 = 36h h = 180 ÷ 36 = 5 So h = 5 cm.

7. Fully worked examples

Example 1: whole-number dimensions

A rectangular prism has length 8 cm, width 3 cm, and height 5 cm. Find its volume.

Step 1: Write the formula. V = l × w × h

Step 2: Substitute values. V = 8 × 3 × 5

Step 3: Multiply. 8 × 3 = 24 24 × 5 = 120

Answer: V = 120 cm^3

Example 2: using base area first

A storage box has a base measuring 12 m by 7 m and a height of 4 m. Find its volume using V = B × h.

Step 1: Find base area. B = l × w = 12 × 7 = 84 m^2

Step 2: Use the volume formula. V = B × h V = 84 × 4

Step 3: Multiply. 84 × 4 = 336

Answer: V = 336 m^3

Example 3: decimal dimensions

A fish tank is 1.2 m long, 0.5 m wide, and 0.4 m high. Find its volume.

Step 1: Write the formula. V = lwh

Step 2: Substitute values. V = 1.2 × 0.5 × 0.4

Step 3: Multiply carefully. 1.2 × 0.5 = 0.6 0.6 × 0.4 = 0.24

Answer: V = 0.24 m^3

Reasonableness check: Since each dimension is less than 2 m, a volume less than 1 m^3 is reasonable.

Example 4: find a missing dimension

A rectangular prism has volume 270 cm^3. Its length is 9 cm and width is 6 cm. Find the height.

Step 1: Start with the formula. V = lwh

Step 2: Substitute known values. 270 = 9 × 6 × h

Step 3: Simplify the known product. 270 = 54h

Step 4: Solve for h. h = 270 ÷ 54 h = 5

Answer: h = 5 cm

Check: 9 × 6 × 5 = 270, so the result is correct.

Example 5: multi-step with unit conversion

A rectangular prism measures 50 cm by 40 cm by 2 m. Find its volume in cm^3.

Step 1: Convert to the same unit. 2 m = 200 cm

Now the dimensions are 50 cm, 40 cm, and 200 cm.

Step 2: Write the formula. V = lwh

Step 3: Substitute. V = 50 × 40 × 200

Step 4: Multiply. 50 × 40 = 2000 2000 × 200 = 400000

Answer: V = 400000 cm^3

Important lesson: Do not multiply mixed units directly. Convert first.

8. Common mistakes to watch for

For a fuller treatment, see rea.m08.geometry-measurement.volume.basic-formulas.misconceptions. The most important checks here are:

Mistake 1: adding dimensions instead of multiplying

Wrong: V = l + w + h

Repair: Volume counts cubes in layers, so multiplication is required: V = l × w × h.

Mistake 2: using square units for volume

Wrong: 96 cm^2

Repair: Volume is measured in cubic units: 96 cm^3.

Mistake 3: forgetting unit conversion

Wrong approach: multiplying 50 cm × 40 cm × 2 m directly.

Repair: Convert all measurements to the same unit first.

Mistake 4: dividing incorrectly in missing-dimension problems

Repair: When solving for one dimension, divide the volume by the product of the other two dimensions.

9. Practice

Try these on your own before checking the answers.

1. A prism has dimensions 7 cm, 4 cm, and 3 cm. Find the volume.
2. A box has base area 30 m^2 and height 9 m. Find the volume.
3. A rectangular prism has volume 192 cm^3, length 8 cm, and width 4 cm. Find the height.
4. A prism measures 2.5 m, 1.2 m, and 0.8 m. Find the volume.
5. A container measures 60 cm by 25 cm by 1.5 m. Find the volume in cm^3.

10. Practice answers

1. V = 7 × 4 × 3 = 84 cm^3
2. V = 30 × 9 = 270 m^3
3. 192 = 8 × 4 × h, so 192 = 32h, therefore h = 6 cm
4. V = 2.5 × 1.2 × 0.8 = 2.4 m^3
5. 1.5 m = 150 cm, so V = 60 × 25 × 150 = 225000 cm^3

11. Summary

For any rectangular prism, volume is the number of cubic units needed to fill it.

• If you know the three dimensions, use V = lwh.
• If you know base area and height, use V = Bh.
• Keep units consistent.
• Write final answers in cubic units.

This lesson handles the rectangular prism formula in full detail. For the wider Grade 8 unit on choosing among volume formulas, use rea.m08.geometry-measurement.volume.basic-formulas.lesson, and for unit-level organization of rectangular prisms specifically, use rea.m08.geometry-measurement.volume.basic-formulas.rectangular-prisms.