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### Course: 5th grade (Eureka Math/EngageNY)>Unit 5

Lesson 1: Topic A: Concepts of volume

# Volume formula intuition

Practice problems that help you see why the volume formula makes sense.

## Volume formula intuition

Problem 1
The rectangular prism has
layers separated by color.
Each colored layer is made up of
unit cubes.
What is the volume of the rectangular prism?
cubic units
A rectangular prism made of unit cubes. The front face of the rectangular prism has 4 red cubes in its bottom layer, 4 yellow cubes in its middle layer, and 4 blue cubes in its top layer. The top face of the rectangular prism has 8 blue cubes arranged in 2 rows and 4 columns. The right-hand side face of the rectangular prism has 2 red cubes in its bottom layer, 2 yellow cubes in its middle layer, and 2 blue cubes in its top layer.

Problem 2
The rectangular prism has
slices separated by color.
Each colored slice is made up of
unit cubes.
What is the volume of the rectangular prism?
cubic units
A rectangular prism made of unit cubes. The front face of the rectangular prism has 12 cubes: 6 rows, 2 columns. The left slice contains 6 green cubes and the right slice contains 6 purple cubes. The top face of the rectangular prism has 10 cubes: 5 rows, 2 columns. The left slice contains 5 green cubes and the right slice contains 5 purple cubes. The right-hand side face of the rectangular prism has 30 purple cubes: 6 rows, 5 columns.

Problem 3
The rectangular prism has
slices separated by color.
Each colored slice is made up of
unit cubes.
What is the volume of the rectangular prism?
cubic units
A rectangular prism made up of unit cubes. The front face of the rectangular prism has 10 cubes: 2 rows and 5 columns. Starting from left to right, the colors of 2 cubes in each column are: blue, pink, green, purple, and red. The top face of the rectangular prism has 20 cubes: 4 rows and 5 columns. Starting from left to right, the colors of 4 cubes in each column are: blue, pink, green, purple, and red. The right hand side face of the rectangular prism has 8 red cubes: 2 rows and 4 columns.

Problem 4
The rectangular prism has
layers separated by color.
Each colored layer is made up of
unit cubes.
What is the volume of the rectangular prism?
cubic units
A rectangular prism made up of unit cubes. The front face of the rectangular prism has 25 cubes: 5 rows and 5 columns. Starting from bottom to top their colors are: 5 cubes are red, 5 cubes are blue, 5 cubes are pink, 5 cubes are green, and 5 cubes are purple. The top face of the rectangular prism has 10 purple cubes: 2 rows and 5 columns. The right hand side face of the rectangular prism has 10 cubes: 5 rows and 2 columns. Starting from bottom to top their colors are: 2 cubes are red, 2 cubes are blue, 2 cubes are pink, 2 cubes are green, and 2 cubes are purple

## Want to join the conversation?

• that was much easier than i thought and i hope it was for other people!
• Your right! That was so easy
• This get VERY easy the more you do it. I encourage everyone to keep going and reach your dream. Remember, NEVER Give Up.
• what does a 4d shape look like
• Interesting question!
An example of a 4d figure is a tesseract, which is a 4d extension of a cube. This can be constructed by drawing two cubes of the same size, then joining each corner (or vertex) of the first cube with the corresponding corner of the second cube (using a line segment to join them).
• it gets super easy and it very fun
• assignment is completed however there is no submit button to turn in the work
• what dose a 4d shape look like
• For example, to draw a representation of a 4d cube (called a tesseract):

1) Draw two 3d cubes, oriented in the same way.
2) Draw a line segment connecting each vertex of one cube to the corresponding vertex of the other cube.

The link https://mathworld.wolfram.com/Tesseract.html shows a picture of a tesseract.

Have a blessed, wonderful day!