Apply Volume Formulas to Prisms

Help Questions

5th Grade Math › Apply Volume Formulas to Prisms

Questions 1 - 10
1

A right rectangular prism has length 9 units, width 2 units, and height 5 units. The base is the 9-by-2 rectangle, and the height is 5 units. Packing 1-unit cubes makes layers, and each layer has the same number of cubes as the base area. How does the base area help find the volume?

Find the base area $9\times2$ and then double it to get volume: $V=2\times(9\times2)$.

Find the base area $9\times2$ and add the height 5: $V=(9\times2)+5$.

Find the base area $9\times2$ and multiply by the height 5: $V=(9\times2)\times5$.

Find the base area by adding $9+2$ and multiply by the height 5: $V=(9+2)\times5$.

Explanation

Volume formulas are used to find the volume of a rectangular prism by quantifying its internal capacity in cubic units. The dimensions represent length as 9 units along one base side, width as 2 units along the other, and height as 5 units upward. This connects to cube layers since each layer holds as many cubes as the base area, with height determining the layer count. The base area of 9 × 2 is multiplied by height 5 to compute the total volume effectively. A misconception is adding dimensions instead of multiplying, which doesn't account for the space filled. These formulas are efficient for rapid results in real-world applications like storage. They generalize across shapes, simplifying volume problems in math and beyond.

2

A right rectangular prism is a cereal box with length 10 cm, width 4 cm, and height 6 cm. The base is the 10 cm by 4 cm rectangle, and the height is 6 cm. The volume should match the number of 1-cm cubes that pack the box. What is the volume of the prism?

20 cubic centimeters

120 square centimeters

40 cubic centimeters

240 cubic centimeters

Explanation

The core skill is using volume formulas to find the volume of a rectangular prism. The length, width, and height represent the spatial extents, with length and width outlining the base and height providing the depth. The formula connects to unit cube layers, filling the base with length × width cubes and stacking height-many layers. Volume is base area × height, matching the packed cubes. A misconception is using square units for volume instead of cubic, but volume always uses cubic units for three dimensions. Formulas are efficient for swift answers in practical situations like packaging. This makes them versatile, and for this cereal box, the volume is 10 × 4 × 6 = 240 cubic centimeters, so the answer is B.

3

A right rectangular prism has length 5 units, width 5 units, and height 6 units. The base is the 5-by-5 square, and the height is 6 units. A student says the formulas for volume represent the same space as packing 1-unit cubes. Which formula correctly finds the volume using base area times height?

Use $V = 2(5\cdot5 + 5\cdot6 + 5\cdot6)$.

Use $V = (5 \times 5) \times 6$.

Use $V = (5+5) \times 6$.

Use $V = 5 \times 6$.

Explanation

Volume formulas are used to find the volume of a rectangular prism by determining its space in cubic units without counting cubes manually. For a square base, length and width are both 5 units, with height 6 units as the vertical measure. The formula links to cube layers by stacking base layers along the height. Base area × height, (5 × 5) × 6, correctly computes the volume. Misconceiving it as surface area formula leads to wrong results like adding faces. Formulas are efficient, saving time over manual methods. They generalize to all prisms, enhancing mathematical understanding.

4

A right rectangular prism has length 10 units, width 3 units, and height 2 units. The base is the 10-by-3 rectangle, and the height is 2 units. Packing cubes would make 2 layers of cubes, each layer matching the base area. What is the volume of the prism?

15 cubic units

112 cubic units

30 cubic units

60 cubic units

Explanation

Volume formulas are used to find the volume of a rectangular prism by calculating its cubic capacity efficiently. Length 10 units and width 3 units form the base, with height 2 units indicating the depth. It connects to cube layers as the base area determines cubes per layer, and height sets the number of layers, resulting in 60 cubic units. Base area × height multiplies 10 × 3 by 2 for the total volume. A misconception is halving instead of multiplying correctly, leading to errors like 15 or 30. These formulas are efficient for quick computations without physical models. They generalize, making volume accessible for everyday uses like packing.

5

A classroom supply bin is a right rectangular prism with length 6 in, width 4 in, and height 7 in. The base is the 6 in by 4 in rectangle, and the height is 7 in. Using $V=b\times h$ gives the same volume as counting how many 1-inch cubes fit inside. Which formula correctly finds the volume?

$V=6\times7$

$V=2(6\times4)+2(6\times7)+2(4\times7)$

$V=(6\times4)\times7$

$V=6+4+7$

Explanation

Volume formulas are used to find the volume of a rectangular prism by multiplying its length, width, and height. Each dimension represents a measurement along one edge: length is the longest side of the base, width is the shorter side, and height is the vertical measurement. The formula connects to filling the prism with layers of 1-unit cubes, where each layer matches the base area. By finding the base area (length × width) and multiplying by height, you get the total number of cubes, or volume. One misconception is thinking volume is the sum of the dimensions, but it's actually their product. Formulas are efficient as they provide a fast way to compute space without building models. They generalize to various applications, like supply bins, where the correct formula is (6 × 4) × 7, making A the right choice.

6

A right rectangular prism is a gift box with length 9 cm, width 2 cm, and height 7 cm. The base is the 9 cm by 2 cm rectangle, and the height is 7 cm. The volume formula gives the same result as packing the box with 1-cm cubes. How does the base area help find volume?

Find base area $9+2$, then add height 7 to get the volume.

Find base area $2\times7$, then multiply by length 9 to get the volume.

Find base area $9\times7$, then multiply by width 2 to get the volume.

Find base area $9\times2$, then multiply by height 7 to get the volume.

Explanation

The core skill is using volume formulas to find the volume of a rectangular prism. The length, width, and height represent the prism's dimensions, where length and width form the base rectangle, and height is the perpendicular measurement upward. The formula relates to layers of 1-cm cubes, with each layer matching the base and the height indicating how many layers stack up. Volume is thus base area (length × width) times height, capturing the total space. A misconception is thinking volume comes from adding dimensions or misidentifying the base, but it requires multiplying the correct base area by height. Formulas are efficient for rapid calculations in everyday scenarios without building models. This approach shows how base area helps, as in finding 9 × 2 then multiplying by 7, which is choice A.

7

A small aquarium is a right rectangular prism with length 7 ft, width 3 ft, and height 4 ft. The base is the 7 ft by 3 ft rectangle, and the height is 4 ft. Filling it with 1-foot cubes would give the same volume as the formula. What is the volume of the prism?

84 cubic feet

70 cubic feet

94 square feet

28 cubic feet

Explanation

Volume formulas are used to find the volume of a rectangular prism, calculating the total internal space. The length, width, and height each represent one of the three perpendicular measurements. Connecting to cube layers, the volume is the base layer's cubes multiplied by the height in layers. Base area × height provides a straightforward way to get this total without gaps. A misconception is using square units for volume, but it must be cubic units. Formulas are efficient because they avoid manual counting for large objects. They apply universally, as in this aquarium where 7 ft × 3 ft × 4 ft = 84 cubic feet, confirming B.

8

A right rectangular prism is a small aquarium with length 12 in, width 5 in, and height 2 in. The base is the 12 in by 5 in rectangle, and the height is 2 in. The formula for volume represents the same space as packing it with 1-inch cubes. Which formula correctly finds the volume?

Add $12+5+2$ to get the volume.

Multiply $12\times5\times2$ to get the volume.

Multiply $12\times5$ to get the volume.

Multiply $2(12+5+2)$ to get the volume.

Explanation

The core skill is using volume formulas to find the volume of a rectangular prism. The length, width, and height represent the defining measurements, with length and width as base dimensions and height as the elevation. The formula links to cube layers, where each base layer has length × width unit cubes, repeated for the height. Thus, volume is base area multiplied by height for total capacity. A misconception is confusing volume with two-dimensional area by omitting one dimension, but all three must be multiplied. Formulas are efficient, bypassing tedious counting for fast results. They extend to various applications, like aquariums, where the correct formula is 12 × 5 × 2, choice C.

9

A storage box is a right rectangular prism with length 8 cm, width 5 cm, and height 3 cm. The base is the 8 cm by 5 cm rectangle, and the height is 3 cm. Packing 1-cm cubes into the box would fill the same space as the volume formula. What is the volume of the prism?

120 cubic centimeters

16 cubic centimeters

40 cubic centimeters

158 square centimeters

Explanation

Volume formulas are used to find the volume of a rectangular prism by multiplying its three dimensions. In a prism, the length and width define the base, while the height determines how tall it is. You can think of the volume as the number of 1-unit cubes that fit into layers stacked along the height. Calculating the base area by multiplying length times width, then multiplying by height, gives the total volume in cubic units. A common misconception is confusing volume with surface area, but volume measures the space inside, not the outer surfaces. These formulas are efficient because they allow quick calculations without physically counting cubes. They also apply to prisms of any size, making them useful for real-world objects like storage boxes, where the volume is 8 cm × 5 cm × 3 cm = 120 cubic centimeters, so the answer is D.

10

A shipping crate is a right rectangular prism with length 5 m, width 4 m, and height 6 m. The base is the 5 m by 4 m face, and the height is 6 m. The volume formula gives the same result as packing 1-meter cubes. Which formula correctly finds the volume?

$V=(5\times4)\times6$

$V=2(5\times4)+2(5\times6)+2(4\times6)$

$V=5\times4+6$

$V=5\times6$

Explanation

Volume formulas are used to find the volume of a rectangular prism by combining its dimensions multiplicatively. Length and width form the base, with height indicating the depth or stacking. The concept ties to cube layers, where each layer holds base area cubes, and height gives the layer count. Thus, base area × height equals the total volume in cubic meters here. Confusing volume with surface area is common, but surface measures outsides, not insides. These formulas are efficient for scaling to big items like crates. They generalize effectively, identifying C as the correct formula for this shipping crate.

Page 1 of 3