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Volume of rectangular prisms made of unit cubes

Volume tells you how much space is inside a 3D object. We can figure it out by seeing how many little 'unit cubes' can fit inside a larger shape. It's like building with blocks to measure space! 📦

Do this: Read the concept below, then try the quiz or activity.

Lesson 41 of 216

Concept

Volume is the measure of space inside a three-dimensional figure. When we use unit cubes, we are counting how many of these standard blocks fit into a rectangular prism (a box shape).

The Two Main Formulas:

1. V = l x w x h
   (Volume = length times width times height)

2. V = B x h
   (Volume = Area of the Base times height)

Both formulas do the same thing! The "Area of the Base" (B) is just another way of saying length x width.

Visualizing the Concept:

Imagine a rectangular prism made of little cubes.
*   Length: How many cubes long is it? (Let's say 5)
*   Width: How many cubes wide is it? (Let's say 3)
*   Height: How many layers of cubes tall is it? (Let's say 4)

1.  Find the Area of the Base (B):
    *   The base is the bottom layer of cubes.
    *   B = length x width = 5 x 3 = 15 cubes on the bottom layer.

2.  Multiply by the Height (h):
    *   Now, stack that layer up. Since the height is 4, you have 4 layers of 15 cubes.
    *   V = B x h = 15 x 4 = 60 cubes.

So, the volume is 60 cubic units.

Using the other formula gives the same result: V = l x w x h = 5 x 3 x 4 = 60 cubic units.

Key Idea: Volume is just counting cubes in a 3D space. The formulas are shortcuts so you don't have to count every single one.

Try it

Practice finding the volume!

Using the Formula (V = l x w x h): 1. A prism is 6 units long, 2 units wide, and 4 units high. What is its volume? 2. A box has a length of 10, a width of 5, and a height of 3. Find the volume. 3. What is the volume of a cube where every side is 4 units? (l=4, w=4, h=4)

Using the Base Area (V = B x h): 4. The base of a prism is a rectangle with an area of 20 square units. If the prism is 5 units high, what is its volume? 5. A prism has a base of 30 square units and a height of 10 units. Find the volume.

Word Problems: 6. A fish tank is 20 inches long, 10 inches wide, and 12 inches high. How many cubic inches of water can it hold? 7. A moving box measures 3 feet long, 2 feet wide, and 2 feet high. What is its volume in cubic feet? 8. You are building a tower with blocks. The base is 8 blocks by 6 blocks. If you build it 10 blocks high, how many blocks will you use in total?

Challenge Problems: 9. A rectangular prism has a volume of 120 cubic units. If its length is 10 and its width is 4, what is its height? 10. A box has a volume of 40. Give two possible sets of dimensions (length, width, height) for this box.

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