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Fraction Classroom · 2.6.0 · Student pages

Account for every part of a joined solid

Partition a stepped solid into non-overlapping rectangular prisms, add their volumes and distinguish a shared face from a region counted twice.

Anonymous group code: __________   Date: __________

The top sits above the leftmost 2 cm of the base. The parts share a 2 cm by 3 cm face and have no overlapping interiors; the combined solid occupies 48 cubic cm.PartDimensionsVolumeBase6 × 3 × 2 cm36 cubic cmTop2 × 3 × 2 cm12 cubic cmCombinedInteriors separate48 cubic cm
The top sits above the leftmost 2 cm of the base. The parts share a 2 cm by 3 cm face and have no overlapping interiors; the combined solid occupies 48 cubic cm.

Start with a diagnostic

diagnostic. A 2 cm × 2 cm × 2 cm prism and a 2 cm × 2 cm × 3 cm prism meet along one 2 cm × 2 cm face, with no overlapping interiors. What is their combined volume? Explain whether the shared face removes any cubic centimetres.

Build the idea

model. A base prism is 6 cm long, 3 cm wide and 2 cm high. A top prism is 2 cm long, 3 cm wide and 2 cm high; it sits above the leftmost 2 cm of the base, sharing a 2 cm × 3 cm face. Their interiors do not overlap. What is the volume of the stepped solid?

  1. Sketch the long base and place the shorter prism above its left end. Colour the base and top differently. Mark their shared face and state that no cube-sized region lies inside both parts.
  2. Record the base's three dimensions in one row: 6 × 3 × 2 = 36 cubic cm. Record the top's own dimensions in a second row: 2 × 3 × 2 = 12 cubic cm. Its 2 cm height is additional to the base's height at that end.
  3. Add the volumes, 36 + 12 = 48 cubic cm. The shared 2 cm × 3 cm face is a boundary with area 6 square cm; it has no thickness and does not duplicate a volume to subtract.
  4. Check the coverage of the record: the base row covers the whole lower prism and the top row covers only the raised prism. The right-hand space above the base is empty. The outer 6 cm by 3 cm by 4 cm dimensions describe a containing box, not a solid prism filled everywhere.

Worked example: 48 cubic cm. The base contributes 6 × 3 × 2 = 36 cubic cm and the top contributes 2 × 3 × 2 = 12 cubic cm. Their non-overlapping interiors cover the solid once, so add 36 + 12.

Practise and explain

Try p1–p3 with a partner. Use a drawing and label your units. Try e1 independently before comparing responses.

p1. Support: prisms measuring 3 cm × 2 cm × 1 cm and 2 cm × 2 cm × 1 cm join end-to-end along 2 cm × 1 cm faces, with no overlapping interiors. Enter the total volume in cubic cm and give one record row for each part.

p2. Core: a base is 5 cm × 4 cm × 1 cm. A 2 cm × 4 cm × 2 cm prism sits above its leftmost 2 cm, with no overlapping interiors. Enter the total volume in cubic cm. State the total height at the raised end and explain which height belongs in the top part's calculation.

p3. Challenge: a plan labels region A as a 4 cm × 2 cm × 1 cm prism and region B as a 3 cm × 1 cm × 1 cm prism. These regions overlap in a 2 cm × 1 cm × 1 cm prism. B's remaining 1 cm × 1 cm × 1 cm part extends beyond A. Enter the actual solid's volume in cubic cm, then redraw a partition using all of A and only B's extension.

e1. Independent check: a base is 4 cm × 3 cm × 2 cm. A 2 cm × 3 cm × 1 cm top prism sits above its leftmost 2 cm, with no overlapping interiors. Enter the total volume in cubic cm. Sketch and label the two parts, give both part volumes and explain why their shared face is not subtracted.

Transfer the idea

transfer. For the worked stepped solid, a learner multiplies its outer length 6 cm, width 3 cm and maximum height 4 cm to obtain 72 cubic cm. Identify the exact empty rectangular prism included in that calculation and use it to check the 48-cubic-cm answer.