Free Math Worksheets

- Use the rectangular-prism volume formula for Package A. - A cube has three equal dimensions. - Compare the two volumes.

1. Package A's volume is \(60\times30\times20=36{,}000\,\text{cm}^3\). 2. Package B's volume is \(35\times35\times35=42{,}875\,\text{cm}^3\). 3. Since \(42{,}875>36{,}000\), Package B takes up more space.

Package A has volume \(36{,}000\,\text{cm}^3\). Package B has volume \(42{,}875\,\text{cm}^3\). Package B takes up more space.

- Read the length, width, and height of each box from the diagram. - First find how many cubes fit in one bottom layer. - Then multiply by the number of layers.

1. Box A measures \(3 \times 3 \times 4\), so its volume is \(3\times3\times4=36\) cubic units. 2. Box B measures \(5 \times 2 \times 2\), so its volume is \(5\times2\times2=20\) cubic units. 3. Box C measures \(2 \times 4 \times 3\), so its volume is \(2\times4\times3=24\) cubic units.

A: \(36\) unit cubes; \(36\) cubic units B: \(20\) unit cubes; \(20\) cubic units C: \(24\) unit cubes; \(24\) cubic units

- Count the cubes along the width, depth, and height of each box. - A multiplication equation is faster than counting every cube one at a time.

1. Box 1 is \(3\) cubes wide, \(3\) cubes deep, and \(2\) cubes high. A matching equation is \(3 \times 3 \times 2 = 18\). 2. Box 2 is \(4\) cubes wide, \(2\) cubes deep, and \(2\) cubes high. A matching equation is \(4 \times 2 \times 2 = 16\).

1) \(18\) unit cubes; \(3 \times 3 \times 2 = 18\) 2) \(16\) unit cubes; \(4 \times 2 \times 2 = 16\)

- Both solids are rectangular prisms. - Multiply the three dimensions for each box. - Compare the two volumes.

1. Box a) has volume \(8\times3\times3=72\,\text{cm}^3\). 2. Box b) has volume \(4\times6\times3=72\,\text{cm}^3\). 3. The two boxes have the same volume.

Yes. Both boxes have a volume of \(72\,\text{cm}^3\).

- Find each block's volume separately. - A cube has three equal dimensions. - Order the numerical results.

1. Block A's volume is \(4\times6\times5=120\,\text{cm}^3\). 2. Block B's volume is \(5\times5\times5=125\,\text{cm}^3\). 3. Block C's volume is \(8\times2\times8=128\,\text{cm}^3\). 4. Therefore, the order is A, B, C.

The order is \(A<B<C\).

- Use the volume formula for a rectangular prism. - Relate cubic centimeters to liters. - Subtract the container's capacity from the amount of juice.

1. The container's volume is \(20\times10\times5=1000\,\text{cm}^3\). 2. Since \(1000\,\text{cm}^3=1\,\text{L}\), the container holds \(1\,\text{L}\). 3. The amount that does not fit is \(1.5\,\text{L}-1\,\text{L}=0.5\,\text{L}\). 4. Since \(0.5\,\text{L}=500\,\text{mL}\), \(500\,\text{mL}\) would overflow.

The container holds \(1\,\text{L}\), so \(500\,\text{mL}\) of the \(1.5\,\text{L}\) would overflow.

- Use the volume formula for a rectangular prism. - Convert cubic centimeters to liters. - Add the mass of the empty aquarium to the mass of the water.

1. The aquarium's volume is \(50\times30\times40=60{,}000\,\text{cm}^3\). 2. Since \(1000\,\text{cm}^3=1\,\text{L}\), its volume is \(60\,\text{L}\). 3. The water has a mass of \(60\,\text{kg}\). 4. The total mass is \(60+12=72\,\text{kg}\).

a) The aquarium's volume is \(60\,\text{L}\). b) The filled aquarium has a total mass of \(72\,\text{kg}\).

- Convert centimeters to decimeters. - Use the volume formula for a cube. - Relate cubic decimeters to liters.

1. Convert the edge length: \(60\,\text{cm}=6\,\text{dm}\). 2. The aquarium's volume is \(6\times6\times6=216\,\text{dm}^3\). 3. Since \(1\,\text{dm}^3=1\,\text{L}\), the capacity is \(216\,\text{L}\).

The aquarium's capacity is \(216\,\text{L}\).

- Use the volume formula for a rectangular prism. - For part b, first find the actual water height. - Convert cubic centimeters to liters.

1. The full aquarium's volume is \(60\times30\times40=72{,}000\,\text{cm}^3\). 2. Since \(1000\,\text{cm}^3=1\,\text{L}\), the full capacity is \(72\,\text{L}\). 3. The current water height is \(40-5=35\,\text{cm}\). 4. The current water volume is \(60\times30\times35=63{,}000\,\text{cm}^3=63\,\text{L}\).

a) The aquarium can hold \(72\,\text{L}\). b) It currently contains \(63\,\text{L}\) of water.

- Choose one length unit for all three dimensions. - Using decimeters makes the final conversion to liters direct. - Multiply only after all dimensions use the same unit.

1. Convert all three lengths to decimeters: \(40\,\text{cm}=4\,\text{dm}\), \(5\,\text{dm}=5\,\text{dm}\), and \(250\,\text{mm}=2.5\,\text{dm}\). 2. Find the volume: \(V=4\times5\times2.5=50\,\text{dm}^3\). 3. Since \(1\,\text{dm}^3=1\,\text{L}\), the volume is \(50\,\text{L}\).

The volume is \(50\,\text{L}\).

- Picture filling the rectangular prism one layer at a time with unit cubes. - What does multiplying two edge lengths find first? - How many unit cubes fit on the chosen base? - How many copies of that layer fill the prism?

1. Using a \(12\,\text{cm}\) by \(5\,\text{cm}\) base, \(V=(12\times5)\times4=60\times4=240\,\text{cm}^3\). 2. Using a \(5\,\text{cm}\) by \(4\,\text{cm}\) base, \(V=12\times(5\times4)=12\times20=240\,\text{cm}^3\). 3. In the first calculation, \(12\times5=60\) is the number of unit cubes in one horizontal layer, and there are \(4\) layers. 4. In the second calculation, \(5\times4=20\) is the number of unit cubes in one vertical layer, and there are \(12\) such layers.

a) \(V=(12\times5)\times4=240\,\text{cm}^3\) \(V=12\times(5\times4)=240\,\text{cm}^3\) b) The product inside the parentheses gives the number of unit cubes in one layer based on the chosen face.

- How many cubes are in one layer? - How many times is that layer repeated? - When the prism is turned, which edge becomes its height? - Are any cubes added or removed when the prism is turned?

1. A row is \(7\,\text{cm}\) long, each layer has \(4\) rows, and there are \(5\) layers. The dimensions are \(7\,\text{cm}\), \(4\,\text{cm}\), and \(5\,\text{cm}\). 2. The volume is \(7\times4\times5=140\,\text{cm}^3\). 3. After the prism is turned, each layer contains \(7\times5=35\) cubes, and the prism has \(4\) layers. 4. The new layer calculation gives \(35\times4=140\,\text{cm}^3\). Turning the prism rearranges the same unit cubes, so its volume does not change.

a) The dimensions are \(7\,\text{cm}\), \(4\,\text{cm}\), and \(5\,\text{cm}\). The volume is \(140\,\text{cm}^3\). b) The statement is correct. In the new position, each layer has \(35\) cubes and there are \(4\) layers, so \(35\times4=140\,\text{cm}^3\).

- Find the number of cubes in one layer by multiplying the length and width. - Determine how many layers are stacked. - Think about what happens when the entire top layer is removed.

1. The original prism contains \(5\times2\times3=30\) cubes. 2. The shorter prism contains \(5\times2\times2=20\) cubes. 3. Jordan must remove \(30-20=10\) cubes.

a) The prism contains \(30\) cubes. b) Jordan must remove \(10\) cubes.

- Find the volume of Luke’s prism. - Find the volume of Sophie’s prism. - Compare the two results.

1. Luke’s prism has volume \(6\times5\times4=120\,\text{cm}^3\), so it contains \(120\) unit cubes. 2. Sophie’s prism has volume \(10\times3\times4=120\,\text{cm}^3\), so it also contains \(120\) unit cubes. 3. The volumes are equal, so neither person used more cubes.

They used the same number of cubes. Each prism has volume \(120\,\text{cm}^3\) and contains \(120\) unit cubes.

- First find the number of cubes in one layer. - The second model uses the same total number of cubes. - Use the relationship among length, width, height, and volume. - Find the missing factor in a multiplication equation with the total number of cubes.

1. Each layer contains \(4\times5=20\) cubes. 2. The first model contains \(20\times3=60\) cubes. 3. For the second model, the length and height account for \(6\times2=12\) cubes in each \(1\,\text{cm}\) of width. 4. The width is \(60\div12=5\,\text{cm}\).

a) \(60\) cubes b) \(5\,\text{cm}\)

- Relate liters to cubic decimeters. - Look for three whole-number factors whose product is \(60\). - Convert each dimension separately from decimeters to centimeters.

1. Since \(1\,\text{L}=1\,\text{dm}^3\), the aquarium's volume is \(60\,\text{dm}^3\). 2. One possible set of factors is \(2\times3\times10=60\), so one set of dimensions is \(2\,\text{dm}\times3\,\text{dm}\times10\,\text{dm}\). 3. Another possible set is \(3\times4\times5=60\), so a second set of dimensions is \(3\,\text{dm}\times4\,\text{dm}\times5\,\text{dm}\). 4. Converting the first set using \(1\,\text{dm}=10\,\text{cm}\) gives \(20\,\text{cm}\times30\,\text{cm}\times100\,\text{cm}\).

a) Sample answers are \(2\,\text{dm}\times3\,\text{dm}\times10\,\text{dm}\) and \(3\,\text{dm}\times4\,\text{dm}\times5\,\text{dm}\). b) The first set is \(20\,\text{cm}\times30\,\text{cm}\times100\,\text{cm}\). Answers will vary.

- Find three whole numbers whose product is \(36\). - Think about how many cubes could be in one layer and how many layers would be needed. - Consider a single long row, a flat layer, or a prism with several layers.

1. For a rectangular prism, \(V=l\times w\times h\), so the three dimensions must have a product of \(36\). 2. Three possible factor triples are \(6\times3\times2=36\), \(9\times4\times1=36\), and \(4\times3\times3=36\). 3. Other whole-number factor triples are also possible.

Three possible sets of dimensions are: 1. \(6\,\text{cm}\times3\,\text{cm}\times2\,\text{cm}\) 2. \(9\,\text{cm}\times4\,\text{cm}\times1\,\text{cm}\) 3. \(4\,\text{cm}\times3\,\text{cm}\times3\,\text{cm}\)

- First find the number of cubes in one layer and the number of layers. - If the total number of cubes stays the same but the height decreases, what happens to the number in each layer? - For part c, find two whole numbers whose product is the answer to part b.

1. The original prism contains \(6\times4\times3=72\) cubes. 2. The new prism has \(2\) layers, so each layer contains \(72\div2=36\) cubes. 3. The new length and width must have a product of \(36\). One possible pair is \(6\,\text{cm}\) and \(6\,\text{cm}\), since \(6\times6=36\).

a) \(72\) unit cubes b) \(36\) cubes per layer c) One possible answer is \(6\,\text{cm}\) long and \(6\,\text{cm}\) wide.

- Picture how the cubes are arranged in each prism. - For part b, find how many cubes are in each \(1\,\text{cm}\)-long section. - A cube has equal length, width, and height. - Test the numbers of unit cubes needed for cubes with edge lengths \(2\,\text{cm}\) and \(3\,\text{cm}\).

1. A single row gives dimensions \(12\,\text{cm}\times1\,\text{cm}\times1\,\text{cm}\). 2. A cross section that is \(2\,\text{cm}\) wide and \(2\,\text{cm}\) high contains \(2\times2=4\) cubes for each centimeter of length. Therefore, the length is \(12\div4=3\,\text{cm}\). 3. A cube with edge length \(2\,\text{cm}\) needs \(2\times2\times2=8\) unit cubes, while a cube with edge length \(3\,\text{cm}\) needs \(3\times3\times3=27\). Because \(12\) is between \(8\) and \(27\), no whole-number edge length makes a cube from exactly \(12\) unit cubes.

a) Length \(12\,\text{cm}\), width \(1\,\text{cm}\), height \(1\,\text{cm}\) b) \(3\,\text{cm}\) c) No. A cube with edge length \(2\,\text{cm}\) uses \(8\) unit cubes, and a cube with edge length \(3\,\text{cm}\) uses \(27\) unit cubes.