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Volume of prisms and composite solids

Learning goals

  • Finds volumes of prisms and combines them for composite solids.
  • Identifies the dimensions or measures required by the formula.
  • Uses cubic units correctly.
  • Solves for missing dimensions when enough information is given.
  • Checks the result for geometric reasonableness.

Click problems to add them to your worksheet.

Volume of prisms and composite solids 1
Separate the stepped solid at the step into rectangular-prism Part 1 and rectangular-prism Part 2. If their volumes are \(V_1\) and \(V_2\), which expression gives the volume of the whole solid? Explain. A. \(V_1+V_2\) B. \(V_1-V_2\) C. \(V_1\times V_2\)
Figure for problem 554234

Hints

- Think about how the two parts combine to make the whole. - Ask whether any part of the solid would be counted twice by adding the two part volumes. - Choose the operation that combines the volumes of separate parts into one total.

Solution

1. The two parts fit together to make the whole solid and occupy different regions of space. 2. Volume is additive for such parts, so the whole volume is the sum of the two part volumes. 3. Therefore, \(V_1+V_2\) is the correct expression.

Answer

A. \(V_1+V_2\). The two parts make the whole solid without occupying the same region, so their volumes add.
Volume of prisms and composite solids 2
A composite solid is made from two rectangular-prism parts that do not overlap. One part has volume \(18\,\text{cm}^3\), and the other has volume \(12\,\text{cm}^3\). Should the two volumes be added or subtracted to find the volume of the whole solid? Explain, then find the volume.

Hints

- Think about what each given volume represents in the whole solid. - Does the whole solid contain all of both prism parts? - After choosing the operation, use the two given volumes.

Solution

1. The two prism parts do not overlap, so each part occupies a different region of the whole solid. 2. Both regions belong to the whole solid, so add their volumes. 3. \(18+12=30\).

Answer

Add the volumes. The parts do not overlap, so each part is counted once. The whole solid has volume \(30\,\text{cm}^3\).
Volume of prisms and composite solids 3
Two students build different solids from unit cubes with a volume of \(1\,\text{cm}^3\) each. Lisa builds a three-level stair-step solid. The bottom level contains \(6\) cubes, the middle level contains \(4\) cubes, and the top level contains \(2\) cubes. Max builds a rectangular prism with dimensions \(3\,\text{cm}\), \(2\,\text{cm}\), and \(2\,\text{cm}\). Compare the volumes of the two solids. Does one have a greater volume, or are the volumes equal?

Hints

- How many unit cubes did Lisa use altogether? - Use the dimensions to find how many unit cubes fit in Max's rectangular prism. - Compare the two totals.

Solution

1. Lisa's solid contains \(6+4+2=12\) unit cubes, so its volume is \(12\,\text{cm}^3\). 2. Max's rectangular prism has a volume of \(3\times2\times2=12\,\text{cm}^3\). 3. The two solids have equal volumes.

Answer

The volumes are equal. Each solid has a volume of \(12\,\text{cm}^3\).
Volume of prisms and composite solids 4
A building block is made from two non-overlapping rectangular prisms. The lower prism is \(4\,\text{cm}\) long, \(2\,\text{cm}\) deep, and \(1\,\text{cm}\) high. The upper prism is \(2\,\text{cm}\) long, \(2\,\text{cm}\) deep, and \(1\,\text{cm}\) high. Find the volume of the composite block.
Figure for problem 535596

Hints

- Identify the two rectangular prisms that make up the solid. - Find the volume of each prism from the dimensions in the problem. - Decide how the two non-overlapping volumes combine.

Solution

1. The lower prism has volume \(4\times2\times1=8\,\text{cm}^3\). 2. The upper prism has volume \(2\times2\times1=4\,\text{cm}^3\). 3. The prisms do not overlap, so the total volume is \(8+4=12\,\text{cm}^3\).

Answer

The composite block has volume \(12\,\text{cm}^3\).
Volume of prisms and composite solids 5
A three-level awards platform is made from three non-overlapping rectangular prisms arranged side by side. Each prism is \(10\,\text{cm}\) wide and \(30\,\text{cm}\) deep. Their heights are \(10\,\text{cm}\), \(20\,\text{cm}\), and \(30\,\text{cm}\). Find the platform's total volume.
Figure for problem 535624

Hints

- Treat each level section as a separate rectangular prism. - The three prisms have the same width and depth but different heights. - Add the volumes because the prisms occupy different regions of space.

Solution

1. The three prism volumes are \(10\times30\times10=3000\,\text{cm}^3\), \(10\times30\times20=6000\,\text{cm}^3\), and \(10\times30\times30=9000\,\text{cm}^3\). 2. The prisms do not overlap, so add their volumes: \(3000+6000+9000=18{,}000\,\text{cm}^3\).

Answer

The platform's total volume is \(18{,}000\,\text{cm}^3\).
Volume of prisms and composite solids 6
Find the volume of the T-shaped beam. The horizontal beam is \(60\,\text{cm}\) long, \(20\,\text{cm}\) high, and \(20\,\text{cm}\) deep. The centered vertical support is \(20\,\text{cm}\) wide, \(30\,\text{cm}\) high, and \(20\,\text{cm}\) deep.
Figure for problem 535802

Hints

- Treat the solid as two rectangular prisms. - Both parts have the same depth. - Add the two volumes.

Solution

1. The horizontal beam has volume \(60\times20\times20=24{,}000\,\text{cm}^3\). 2. The vertical support has volume \(20\times30\times20=12{,}000\,\text{cm}^3\). 3. The total volume is \(24{,}000+12{,}000=36{,}000\,\text{cm}^3\).

Answer

The volume is \(36{,}000\,\text{cm}^3\).
Volume of prisms and composite solids 7
Split the L-shaped solid into a left rectangular prism and a right rectangular prism. Find its total volume.
Figure for problem 554235

Hints

- Read the dimensions of the two non-overlapping rectangular parts. - Find each prism's volume separately. - Add the partial volumes.

Solution

1. The left prism has volume \(2\times4\times3=24\,\text{cm}^3\). 2. The right prism has volume \(3\times4\times1=12\,\text{cm}^3\). 3. The two prisms do not overlap, so the total volume is \(24+12=36\,\text{cm}^3\).

Answer

\(36\,\text{cm}^3\)
Volume of prisms and composite solids 8
A science-fair display stand is made from two non-overlapping rectangular-prism sections. The lower section has base area \(30\,\text{cm}^2\) and height \(4\,\text{cm}\). The upper section has base area \(12\,\text{cm}^2\) and height \(5\,\text{cm}\). Find the total volume of the stand.

Hints

- Find each section's volume from its base area and height. - Keep the two component volumes separate until both are known. - Add the volumes because the sections are non-overlapping parts of one solid.

Solution

1. Use \(V=B\times h\) for the lower section: \(30\times4=120\,\text{cm}^3\). 2. Use \(V=B\times h\) for the upper section: \(12\times5=60\,\text{cm}^3\). 3. The sections do not overlap, so add their volumes: \(120+60=180\,\text{cm}^3\).

Answer

\(180\,\text{cm}^3\)
Volume of prisms and composite solids 9
Two rectangular prisms form one solid. Prism A has volume \(6\,\text{cm}^3\), Prism B has volume \(6\,\text{cm}^3\), and the prisms overlap in a region with volume \(2\,\text{cm}^3\). What is the volume of the combined solid? Explain why the overlap is subtracted.

Hints

- What would \(6+6\) count? - Think about whether any region would then be counted more than once. - Make one adjustment so each part of the combined solid is counted once.

Solution

1. Adding the two full prism volumes gives \(6+6=12\,\text{cm}^3\). 2. The \(2\,\text{cm}^3\) overlap was included once in each prism, so it was counted twice. 3. Subtract the overlap once: \(12-2=10\,\text{cm}^3\).

Answer

\(10\,\text{cm}^3\). The overlap is subtracted once because adding the two full prism volumes counts that region twice.
Volume of prisms and composite solids 10
Prism A has volume \(12\,\text{cm}^3\), and Prism B has volume \(9\,\text{cm}^3\). They overlap, and the combined solid has volume \(17\,\text{cm}^3\). What is the volume of the overlap?

Hints

- Find the sum of the two prism volumes if they are counted separately. - Compare that sum with the actual combined volume. - What could make the separate-volume sum larger than the combined volume?

Solution

1. The two full prism volumes add to \(12+9=21\,\text{cm}^3\). 2. The combined solid has volume \(17\,\text{cm}^3\), so \(21-17=4\,\text{cm}^3\) was counted twice in the full-volume sum. 3. The overlap has volume \(4\,\text{cm}^3\).

Answer

\(4\,\text{cm}^3\)
Volume of prisms and composite solids 11
A building is made from unit cubes with edge length \(1\,\text{cm}\). A square shaft runs through the center from the top to the bottom. What is the volume of the building in cubic centimeters?
Figure for problem 531940

Hints

- First imagine the structure as a completely filled large cube. - Find the volume of the empty shaft through the center. - Subtract the shaft's volume from the full cube's volume. - Each unit cube has volume \(1\,\text{cm}^3\).

Solution

1. A solid \(3\times3\times3\) cube would have volume \(3\times3\times3=27\,\text{cm}^3\). 2. The central shaft measures \(1\,\text{cm}\times1\,\text{cm}\times3\,\text{cm}\), so its volume is \(3\,\text{cm}^3\). 3. Subtract the shaft: \(27-3=24\,\text{cm}^3\).

Answer

\(24\,\text{cm}^3\)
Volume of prisms and composite solids 12
A solid metal cube has edge length \(6\,\text{cm}\). A smaller cube with edge length \(2\,\text{cm}\) is cut completely from one corner. Find the volume of the remaining solid.
Figure for problem 535551

Hints

- Find the volume of the original cube. - Find the volume of the removed cube. - Subtract the removed volume from the original volume.

Solution

1. The original cube has volume \(6\times6\times6=216\,\text{cm}^3\). 2. The removed cube has volume \(2\times2\times2=8\,\text{cm}^3\). 3. The remaining volume is \(216-8=208\,\text{cm}^3\).

Answer

The remaining solid has volume \(208\,\text{cm}^3\).
Volume of prisms and composite solids 13
Solid a) is made from two non-overlapping rectangular prisms. The lower prism measures \(6\,\text{cm}\times4\,\text{cm}\times2\,\text{cm}\). The upper prism measures \(2\,\text{cm}\times4\,\text{cm}\times4\,\text{cm}\). Solid b) is a rectangular prism measuring \(10\,\text{cm}\times4\,\text{cm}\times2\,\text{cm}\). Do the two solids have the same volume? Justify your answer.
Figure for problem 535706

Hints

- The two parts of solid a) do not overlap. - Find the volume of solid a) by combining its two rectangular-prism volumes. - Compare that result with the volume of solid b).

Solution

1. Solid a) has volume \(6\times4\times2+2\times4\times4=48+32=80\,\text{cm}^3\). 2. Solid b) has volume \(10\times4\times2=80\,\text{cm}^3\). 3. The two solids have equal volume even though their shapes are different.

Answer

Yes. Both solids have volume \(80\,\text{cm}^3\).
Volume of prisms and composite solids 14
Solid A is a rectangular prism measuring \(10\,\text{cm}\times2\,\text{cm}\times6\,\text{cm}\). Solid B is made from a lower prism measuring \(8\,\text{cm}\times5\,\text{cm}\times2\,\text{cm}\) and an upper prism measuring \(4\,\text{cm}\times5\,\text{cm}\times2\,\text{cm}\). Solid C is made from a lower prism measuring \(12\,\text{cm}\times2\,\text{cm}\times4\,\text{cm}\) and an upper prism measuring \(4\,\text{cm}\times2\,\text{cm}\times4\,\text{cm}\). Find each volume. Which solids have equal volumes?
Figure for problem 535723

Hints

- Use the stated dimensions rather than estimating from the drawing. - Solid A is one rectangular prism; solids B and C each contain two non-overlapping rectangular prisms. - Compare the three calculated totals.

Solution

1. Solid A has volume \(10\times2\times6=120\,\text{cm}^3\). 2. Solid B has volume \(8\times5\times2+4\times5\times2=80+40=120\,\text{cm}^3\). 3. Solid C has volume \(12\times2\times4+4\times2\times4=96+32=128\,\text{cm}^3\). 4. Solids A and B have equal volumes.

Answer

Solid A: \(120\,\text{cm}^3\) Solid B: \(120\,\text{cm}^3\) Solid C: \(128\,\text{cm}^3\) Solids A and B have equal volumes.
Volume of prisms and composite solids 15
An aluminum channel is \(5\,\text{cm}\) wide, \(3\,\text{cm}\) long, and \(4\,\text{cm}\) high. The bottom and both side walls are \(1\,\text{cm}\) thick. Find its volume.
Figure for problem 535792

Hints

- Look for a way to partition the channel into rectangular prisms that cover the solid exactly once. - Use the wall thickness and overall dimensions to infer any piece dimensions you need. - Check that the pieces do not overlap before adding their volumes.

Solution

1. The bottom prism has volume \(5\times3\times1=15\,\text{cm}^3\). 2. Each side wall is \(4-1=3\,\text{cm}\) high and has volume \(1\times3\times3=9\,\text{cm}^3\). 3. The total volume is \(15+2\times9=33\,\text{cm}^3\).

Answer

The volume is \(33\,\text{cm}^3\).
Volume of prisms and composite solids 16
A hollow metal block has outer dimensions \(12\,\text{cm}\times10\,\text{cm}\times6\,\text{cm}\). A rectangular opening measuring \(8\,\text{cm}\times10\,\text{cm}\times4\,\text{cm}\) passes through the block. Find the volume of metal in the block.
Figure for problem 535909

Hints

- Find the volume of the outer rectangular prism. - Subtract the volume of the opening. - Your final volume should be smaller than the volume of the outer prism.

Solution

1. The outer prism has volume \(12\times10\times6=720\,\text{cm}^3\). 2. The opening has volume \(8\times10\times4=320\,\text{cm}^3\). 3. The metal volume is \(720-320=400\,\text{cm}^3\).

Answer

The metal volume is \(400\,\text{cm}^3\).
Volume of prisms and composite solids 17
A solid metal part fits inside a rectangular prism measuring \(32\,\text{cm}\times15\,\text{cm}\times20\,\text{cm}\). A rectangular notch measuring \(12\,\text{cm}\times15\,\text{cm}\times8\,\text{cm}\) is removed from the top center. Find the volume of the remaining metal part.
Figure for problem 536091

Hints

- Think of the solid as one full rectangular prism with a smaller rectangular prism removed. - Find the two prism volumes from the dimensions in the problem. - The missing notch should be subtracted, not added.

Solution

1. The enclosing rectangular prism has volume \(32\times15\times20=9600\,\text{cm}^3\). 2. The removed notch has volume \(12\times15\times8=1440\,\text{cm}^3\). 3. Subtract the removed volume: \(9600-1440=8160\,\text{cm}^3\).

Answer

The remaining volume is \(8160\,\text{cm}^3\).
Volume of prisms and composite solids 18
A composite solid is made from two non-overlapping rectangular prisms and has total volume \(150\,\text{cm}^3\). Prism A measures \(6\,\text{cm}\times5\,\text{cm}\times3\,\text{cm}\). Prism B has a \(5\,\text{cm}\times3\,\text{cm}\) base and an unknown height. Find the height of prism B.

Hints

- First find how much of the total volume belongs to prism A. - Subtract to find prism B's volume. - Then work backward from prism B's base area and volume.

Solution

1. Prism A has volume \(6\times5\times3=90\,\text{cm}^3\). 2. Prism B must have volume \(150-90=60\,\text{cm}^3\). 3. Its base area is \(5\times3=15\,\text{cm}^2\). 4. Since \(60=15\times h\), \(h=4\,\text{cm}\).

Answer

\(4\,\text{cm}\)
Volume of prisms and composite solids 19
A composite solid is made from two non-overlapping rectangular prisms and has volume \(48\,\text{cm}^3\). Prism A has volume \(24\,\text{cm}^3\). Prism B has base area \(8\,\text{cm}^2\). What is the height of Prism B?

Hints

- First find how much of the total volume belongs to Prism B. - Then use Prism B's base area and volume. - Work backward from \(V=B\times h\) to find the height.

Solution

1. Since the prisms do not overlap, Prism B has volume \(48-24=24\,\text{cm}^3\). 2. Use \(V=B\times h\) for Prism B: \(24=8\times h\). 3. \(h=3\,\text{cm}\).

Answer

\(3\,\text{cm}\)
Volume of prisms and composite solids 20
A concrete staircase has three steps. Each step is \(2\,\text{dm}\) high and \(3\,\text{dm}\) deep. The staircase is \(10\,\text{dm}\) wide. Find its volume in cubic decimeters and in liters.
Figure for problem 531583

Hints

- Look for a way to describe the staircase as rectangular-prism parts that do not overlap. - Think about how the step height and depth determine the dimensions of those parts. - What relationship between cubic decimeters and liters can you use for the final conversion?

Solution

1. Split the staircase into three horizontal rectangular-prism layers. 2. Their volumes are \(9\times10\times2=180\,\text{dm}^3\), \(6\times10\times2=120\,\text{dm}^3\), and \(3\times10\times2=60\,\text{dm}^3\). 3. The total volume is \(180+120+60=360\,\text{dm}^3\). 4. Since \(1\,\text{dm}^3=1\,\text{L}\), the staircase uses \(360\,\text{L}\) of concrete.

Answer

The volume is \(360\,\text{dm}^3\), or \(360\,\text{L}\).
Volume of prisms and composite solids 21
A solid is formed by a horizontal rectangular prism measuring \(6\,\text{cm}\times4\,\text{cm}\times2\,\text{cm}\) and a vertical rectangular prism measuring \(2\,\text{cm}\times4\,\text{cm}\times5\,\text{cm}\). The vertical prism passes through the full \(2\,\text{cm}\) height of the horizontal prism. Lena adds the two full prism volumes and gets \(88\,\text{cm}^3\). Is Lena’s method valid for the volume of the solid? Explain your decision and find the correct volume.

Hints

- Before deciding whether two full volumes can simply be added, ask whether the two prisms occupy any of the same space. - Identify the dimensions of any shared region. - If a region was counted twice, adjust the sum so every part of the solid is counted once.

Solution

1. The horizontal prism has volume \(6\times4\times2=48\,\text{cm}^3\). The vertical prism has volume \(2\times4\times5=40\,\text{cm}^3\). 2. Lena’s method is not valid because the two named prisms overlap: the \(2\times4\times2\) portion of the vertical prism lies inside the horizontal prism. Adding both full volumes counts that region twice. 3. The overlap has volume \(2\times4\times2=16\,\text{cm}^3\). 4. Correct the double count: \(48+40-16=72\,\text{cm}^3\).

Answer

Lena’s method is not valid because the two prisms overlap. The overlap is \(16\,\text{cm}^3\), so the correct volume is \(48+40-16=72\,\text{cm}^3\).

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