Calculate Cube Volume from 9 cm² Base Area: Step-by-Step Solution

Q: How can I find volume if I only know one face area?

In a cube, all faces are identical! So if you know one face area is 9 cm², every face has the same area. This gives you enough information to find the side length.

Q: Why do I take the square root of the area?

Because area of a square = $ s^2 $. To find the side length s, you need to "undo" the squaring by taking the square root: $ s = \sqrt{\text{area}} $.

Q: Is there a difference between base area and face area in a cube?

No difference at all! In a cube, the base, top, and all four sides are identical squares. So "base area" just means the area of any face.

Cube Volume with Given Base Area

A cube has a base area of 9 cm².

Is it possible to calculate the volume of the cube? If so, what is it?

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Calculate the volume of the cube if possible

00:04 In a cube all edges are equal, let's mark it with A

00:07 We'll use the formula for calculating base area (height times length)

00:13 We'll extract the root and find the 2 possible solutions

00:17 A must be positive, as it is the length of an edge

00:21 This is the length of the edge in the cube

00:27 In a cube all edges are equal

00:31 We'll use the formula for calculating volume (edge cubed)

00:35 We'll substitute the edge value we found and solve for the volume

00:41 And this is the solution to the problem

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

A cube has a base area of 9 cm².

Is it possible to calculate the volume of the cube? If so, what is it?

Step-by-step solution

To determine if we can calculate the volume of the cube, let's start by analyzing the given information:

The base area of the cube is given as $9 \, \text{cm}^2$ . In a cube, each face is a square, so this area corresponds to the area of one face.
To find the side length $s$ of the square face, use the formula for the area of a square: $A = s^2$ .
Set up the equation based on the given area: $s^2 = 9$ .
Solve for $s$ by taking the square root of both sides: $s = \sqrt{9} = 3 \, \text{cm}$ .
Now that we have the side length $s$ , calculate the volume $V$ of the cube using the formula for the volume of a cube: $V = s^3$ .
Substitute $s = 3 \, \text{cm}$ into the volume formula: $V = 3^3 = 27 \, \text{cm}^3$ .

Therefore, the volume of the cube is $27 \, \text{cm}^3$ .

Among the given choices, the correct answer is:

Choice 3: $27$

Final Answer

$27$

Key Points to Remember

Essential concepts to master this topic

Cube Property: All faces are identical squares with equal areas
Technique: From area 9 cm², find side: $s = \sqrt{9} = 3$ cm
Check: Volume formula $V = s^3 = 3^3 = 27$ cm³ ✓

Common Mistakes

Avoid these frequent errors

Confusing area formula with volume formula
Don't use area = 9 directly as volume or try to cube the area itself = wrong dimensions! Area is cm² but volume needs cm³. Always find the side length first using $s = \sqrt{\text{area}}$ , then calculate volume using $V = s^3$ .

Practice Quiz

Test your knowledge with interactive questions

A cube has a total of 14 edges.

FAQ

Everything you need to know about this question

How can I find volume if I only know one face area?

In a cube, all faces are identical! So if you know one face area is 9 cm², every face has the same area. This gives you enough information to find the side length.

Why do I take the square root of the area?

Because area of a square = $s^2$ . To find the side length s, you need to "undo" the squaring by taking the square root: $s = \sqrt{\text{area}}$ .

Is there a difference between base area and face area in a cube?

No difference at all! In a cube, the base, top, and all four sides are identical squares. So "base area" just means the area of any face.

What if the area doesn't have a perfect square root?

You might get a decimal or need to leave it as a square root. For example, if area = 8 cm², then $s = \sqrt{8} = 2\sqrt{2}$ cm, and volume = $(2\sqrt{2})^3$ .

How do I check my volume answer is correct?

Work backwards! Take the cube root of your volume to get the side length, then square it to get area. If you get back to 9 cm², your answer is right! $\sqrt[3]{27} = 3$ , and $3^2 = 9$ ✓

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