Calculate Square Base Dimensions: 36 cm³ Rectangular Prism with Height 9

Q: Why do we use x² in the volume formula instead of just x?

Because the base is a square! The area of a square with side length x is $ x^2 $. Since volume equals base area × height, we get $ V = x^2 \times h $.

Q: How do I know which dimension is the height?

The problem states "given that its height is 9" - this tells us directly! The height is the vertical dimension that's perpendicular to the square base.

Q: What if I get a negative answer when taking the square root?

In geometry problems, we only use the positive square root because lengths cannot be negative. So $ \sqrt{4} = 2 $, not -2.

Q: Can I solve this problem without using the volume formula?

No, you need the volume formula $ V = \text{base area} \times \text{height} $! This is the fundamental relationship that connects all the given information.

Q: Why is the answer 2 cm and not 4 cm?

We solve $ x^2 = 4 $ by taking the square root: $ x = \sqrt{4} = 2 $. The 4 represents the area of the square base, but each side length is 2 cm.

Volume Formulas with Square Base Prisms

A rectangular prism with a volume of 36 cm³ has a square base.

Calculate the lengths of the sides of the base given that its height is 9.

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Understand the problem

A rectangular prism with a volume of 36 cm³ has a square base.

Calculate the lengths of the sides of the base given that its height is 9.

Step-by-step solution

To solve this problem, we'll follow these steps:

Step 1: Write out the volume formula $V = x^2 \times h$ , where $x$ is the side of the square base and $h$ is the height.
Step 2: Substitute the given values $V = 36 \, \text{cm}^3$ and $h = 9 \, \text{cm}$ into the formula.
Step 3: Solve for $x$ .

Now, let's work through each step:
Step 1: The volume formula for the prism is $V = x^2 \times h$ .
Step 2: Substitute the known values: $36 = x^2 \times 9$ .
Step 3: Solve for $x$ by dividing both sides by 9: $x^2 = \frac{36}{9} = 4$ .
To find $x$ , take the square root of both sides: $x = \sqrt{4} = 2$ .

Therefore, the length of each side of the square base is $x = 2 \, \text{cm}$ .

Final Answer

Key Points to Remember

Essential concepts to master this topic

Formula: Volume of square base prism equals base area times height
Technique: Substitute known values: $36 = x^2 \times 9$
Check: Verify by calculating: $2^2 \times 9 = 4 \times 9 = 36$ ✓

Common Mistakes

Avoid these frequent errors

Confusing height with base side length
Don't use height (9 cm) as the base side length = wrong answer of 4 cm²! This mixes up the dimensions and ignores the square base property. Always identify which measurement is the height and solve for the unknown base dimension using $V = x^2 \times h$ .

Practice Quiz

Test your knowledge with interactive questions

Calculate the volume of the rectangular prism below using the data provided.

FAQ

Everything you need to know about this question

Why do we use x² in the volume formula instead of just x?

Because the base is a square! The area of a square with side length x is $x^2$ . Since volume equals base area × height, we get $V = x^2 \times h$ .

How do I know which dimension is the height?

The problem states "given that its height is 9" - this tells us directly! The height is the vertical dimension that's perpendicular to the square base.

What if I get a negative answer when taking the square root?

In geometry problems, we only use the positive square root because lengths cannot be negative. So $\sqrt{4} = 2$ , not -2.

Can I solve this problem without using the volume formula?

No, you need the volume formula $V = \text{base area} \times \text{height}$ ! This is the fundamental relationship that connects all the given information.

Why is the answer 2 cm and not 4 cm?

We solve $x^2 = 4$ by taking the square root: $x = \sqrt{4} = 2$ . The 4 represents the area of the square base, but each side length is 2 cm.

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