Volume of Rectangular Prisms
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Learning Objectives
- 1Compute the volume of a rectangular prism using V = l x w x h in cubic inches, cubic feet, cubic cm, or cubic meters
- 2Compute the volume of a cube using V = s x s x s in either US customary or metric units
- 3Express volume in cubic units (cu in, cu ft, cm^3, m^3) and distinguish it from area and perimeter
- 4Solve real-world volume word problems with US customary and metric measurements
Mini Lesson
Volume is the amount of 3D space INSIDE a solid shape. While area covers a flat surface and perimeter measures the outline, volume measures how much fits inside a box, a tank, or any container. Volume is always measured in CUBIC units — three dimensions multiplied together. US customary cubic units are cu in (cubic inches), cu ft (cubic feet), and cu yd (cubic yards). Metric cubic units are cm^3 (cubic centimeters) and m^3 (cubic meters). The formula V = l x w x h is the same in either system; only the unit labels change.
Rectangular Prisms
- A rectangular prism is a box-shaped solid with six rectangular faces.
- It has three measurements: length (l), width (w), and height (h).
- Volume formula: V = l x w x h. Multiply all three to find how many unit cubes fit inside.
- Always include cubic units in the answer: 30 cu in or 30 cu ft (customary); 30 cm^3 or 30 m^3 (metric).
- Keep units consistent inside one problem — do not multiply inches by centimeters.
Parallel Worked Examples: Box 5 x 3 x 2
- Customary — l = 5 in, w = 3 in, h = 2 in. V = 5 x 3 x 2 = 30 cu in. Thirty inch-cubes fit inside.
- Metric — l = 5 cm, w = 3 cm, h = 2 cm. V = 5 x 3 x 2 = 30 cm^3. Thirty cm-cubes fit inside.
- First multiply 5 x 3 = 15. Then 15 x 2 = 30. The arithmetic is identical; only the unit changes.
- Larger version: a box 5 ft x 3 ft x 2 ft has V = 30 cu ft, and the same shape in meters has V = 30 m^3.
Cubes
- A cube is a special rectangular prism where every edge has the same length s.
- Volume formula: V = s x s x s, also written V = s^3 (read "s cubed").
- Customary example: a cube with side 4 in has V = 4 x 4 x 4 = 64 cu in.
- Metric example: a cube with side 4 cm has V = 64 cm^3.
- The exponent 3 is why volume uses CUBIC units — three dimensions multiplied.
Volume vs Area vs Perimeter
- Perimeter is 1D: distance around a flat shape (units: in, ft, cm, m).
- Area is 2D: flat surface inside a shape (units: sq in, sq ft, cm^2, m^2 — square).
- Volume is 3D: space inside a solid (units: cu in, cu ft, cm^3, m^3 — cubic).
- Match the unit to the dimension. If the question asks "how much fits inside," it is volume.
Tip: when computing V = l x w x h, you can multiply in any order. 5 x 3 x 2 = 3 x 2 x 5 = 30. Pick the order that gives easy products first. Always label your answer with the matching cubic unit — cu in, cu ft, cm^3, or m^3.
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