Grade 5

Elementary

7 min

Lesson 10 of 12

The Volume Formula: Length, Width, Height

Not Started

Move beyond counting cubes and use the mathematical formula to find volume quickly.

Imagine you are a professional gift wrapper. You have a giant box to fill with smaller presents—do you have to count every single one, or is there a 'math shortcut' to find the space inside instantly?

Learning Objectives

1.Apply the formula $V = L \times W \times H$ to find the volume of rectangular prisms.
2.Use the alternative formula $V = \text{Base Area} \times H$ when the bottom surface is already known.
3.Break down complex, 'L-shaped' composite shapes into simpler boxes to calculate total volume.

The Power of Three

In earlier grades, you counted unit cubes one by one. But what if the box is huge? We use Volume, which is the amount of 3D space an object occupies. For a rectangular prism, we multiply three dimensions: Length (

l

), Width (

w

), and Height (

h

). Think of it like this: the Length and Width tell you how many cubes are on the bottom floor, and the Height tells you how many floors high the building is! The formula is:

V = L \times W \times H

Volume is always measured in cubic units, like

cm^3

in^3

, because we are multiplying three measurements together.

Find the volume of a shoebox that is $10$ inches long, $5$ inches wide, and $4$ inches tall.

1. Identify the dimensions: $L = 10$ , $W = 5$ , $H = 4$ . 2. Write the formula: $V = L \times W \times H$ . 3. Substitute the numbers: $V = 10 \times 5 \times 4$ . 4. Multiply: $10 \times 5 = 50$ . Then $50 \times 4 = 200$ . 5. Add units: $200 \text{ in}^3$ .

Quick Check

If a box has a length of $2$ cm, a width of $3$ cm, and a height of $10$ cm, what is its volume?

Multiply

2 \times 3 \times 10

Answer

$60 \text{ cm}^3$

The Base Area Shortcut

Sometimes, you already know the area of the bottom of the box. This is called the Base Area (

B

). Since Area is just

L \times W

, we can simplify our formula! If you know the area of the floor, you just multiply it by the height to 'stack' that area all the way to the top. This gives us the formula:

V = B \times H

This is very helpful when looking at shapes like swimming pools or storage units where the floor size is already measured for you.

A small fish tank has a Base Area of $50 \text{ cm}^2$ . If the water is $12 \text{ cm}$ deep, what is the volume of the water?

1. Identify the Base Area ( $B = 50$ ) and Height ( $H = 12$ ). 2. Use the formula: $V = B \times H$ . 3. Calculate: $50 \times 12$ . 4. Mental math tip: $5 \times 12 = 60$ , then add the zero from the $50$ . 5. Result: $600 \text{ cm}^3$ .

Quick Check

If the Base Area of a box is $20 \text{ sq ft}$ and the height is $5 \text{ ft}$ , what is the volume?

Multiply the area of the floor by the height.

Answer

$100 \text{ cubic feet}$

Advanced Application: Composite Shapes

What happens if the shape isn't a perfect box? Imagine an 'L-shaped' building. We call these composite shapes. To find their volume, we use a 'Slice and Add' strategy. First, slice the shape into two or more smaller rectangular prisms. Second, calculate the volume of each piece separately using $L \times W \times H$ . Finally, add the volumes together to get the total. It’s like finding the volume of two separate rooms and adding them to find the volume of the whole house!

An L-shaped step is made of two blocks. Block A is $2 \times 2 \times 3$ . Block B is $4 \times 2 \times 1$ . What is the total volume?

1. Calculate Volume A: $V_A = 2 \times 2 \times 3 = 12$ . 2. Calculate Volume B: $V_B = 4 \times 2 \times 1 = 8$ . 3. Add them together: $12 + 8 = 20$ . 4. Final Answer: $20 \text{ cubic units}$ .

Key Takeaways

1Volume is the total 3D space inside an object, measured in cubic units.
2The standard formula is $V = \text{Length} \times \text{Width} \times \text{Height}$ .
3The shortcut formula is $V = \text{Base Area} \times \text{Height}$ .
4For complex shapes, split them into smaller boxes, find each volume, and add them up.

Quick Check

Multiple Choice

A box is $4$ m long, $3$ m wide, and $2$ m high. What is its volume?

Multiple Choice

If $V = B \times H$ , and the volume is $30 \text{ in}^3$ with a height of $5 \text{ in}$ , what is the Base Area?

True/False

To find the volume of a composite shape, you should subtract the smaller volumes from the larger ones.

What's Next?

Review Tomorrow

In 24 hours, try to recall the two different formulas for volume and explain to someone why we use 'cubic' units.

Practice Activity

Find a cereal box or a tissue box at home. Measure its length, width, and height in centimeters and calculate how much space is inside!

Browse

Browse

The Volume Formula: Length, Width, Height

Learning Objectives

The Power of Three

The Base Area Shortcut

Advanced Application: Composite Shapes

Key Takeaways

Quick Check

What's Next?

The Volume Formula: Length, Width, Height

Learning Objectives

The Power of Three

The Base Area Shortcut

Advanced Application: Composite Shapes

Key Takeaways

Quick Check

What's Next?