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

A rectangular prism has a base measuring 5 units by 8 units.

The height of the prism is 12 units.

Calculate its volume.

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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