Find Rectangle Dimensions: Area 256 cm² with 4:1 Side Ratio

Q: Why do we get x² in the equation?

When you multiply x by 4x, you get $ 4x^2 $. This happens because area involves multiplying two dimensions, and when both dimensions contain the variable x, you naturally get x².

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

Always take the positive square root when dealing with measurements! Lengths can't be negative in real-world problems. So $ \sqrt{64} = 8 $, not -8.

Q: How can I check my answer makes sense?

Verify three things: (1) Your dimensions multiply to give the correct area, (2) One dimension is exactly 4 times the other, and (3) Both dimensions are positive numbers.

Rectangle Area Problems with Side Ratios

The area of a rectangle is 256 cm².

One side is 4 times longer than the other.

What are the dimensions of the rectangle?

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Determine the rectangle's dimensions

00:04 Mark one side as the length X

00:07 According to the given data, the second side is 4 times larger than the first side

00:10 Apply the formula for calculating the area of a rectangle

00:13 Side(X) x side (4X)

00:16 Substitute in the relevant values and proceed to solve for X

00:24 Take the square root

00:29 That's the value of X, now let's substitute in the rectangle's sides to determine the dimensions

00:35 Substitute in the side that's 4 times larger

00:41 That's the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

The area of a rectangle is 256 cm².

One side is 4 times longer than the other.

What are the dimensions of the rectangle?

Step-by-step solution

To find the area of the rectangle, we multiply the length by the width.

According to the data in the statement, one side will be equal to X and the other side will be equal to 4X

Now we replace the existing data:

$S=x\times4x$

$256=4x^2$

We divide the two sections by 4:

$64=x^2$

We extract the square root:

$x=\sqrt{64}=8$

If we said that one side is equal to x and the other side is equal to 4x and we know that x=8

From here we can conclude that the sides of the rectangle are equal:

$8,8\times4=8,32$

Final Answer

8 x 32

Key Points to Remember

Essential concepts to master this topic

Area Formula: Rectangle area equals length times width (A = l × w)
Variable Setup: Let shorter side = x, then longer side = 4x
Verification: Check that 8 × 32 = 256 and 32 ÷ 8 = 4 ✓

Common Mistakes

Avoid these frequent errors

Setting up the equation incorrectly with ratios
Don't write the equation as x + 4x = 256 (adding instead of multiplying)! This treats the problem like perimeter instead of area. Always remember that area = length × width, so write $x \times 4x = 256$ .

Practice Quiz

Test your knowledge with interactive questions

Look at the rectangle below.

Side AB is 2 cm long and side BC has a length of 7 cm.

What is the perimeter of the rectangle?

FAQ

Everything you need to know about this question

How do I know which side to call x and which to call 4x?

It doesn't matter! You can call either side x. If the shorter side is x, then the longer side is 4x. If you accidentally call the longer side x, you'll get x = 32 and the other side would be x/4 = 8, giving the same dimensions.

Why do we get x² in the equation?

When you multiply x by 4x, you get $4x^2$ . This happens because area involves multiplying two dimensions, and when both dimensions contain the variable x, you naturally get x².

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

Always take the positive square root when dealing with measurements! Lengths can't be negative in real-world problems. So $\sqrt{64} = 8$ , not -8.

How can I check my answer makes sense?

Verify three things: (1) Your dimensions multiply to give the correct area, (2) One dimension is exactly 4 times the other, and (3) Both dimensions are positive numbers.

What if the ratio was different, like 3:1 or 5:2?

The same method works! For a 3:1 ratio, use x and 3x. For a 5:2 ratio, use 2x and 5x. The key is to represent both sides using the same variable in the correct proportion.

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Find Rectangle Dimensions: Area 256 cm² with 4:1 Side Ratio

Rectangle Area Problems with Side Ratios

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

How do I know which side to call x and which to call 4x?

Why do we get x² in the equation?

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

How can I check my answer makes sense?

What if the ratio was different, like 3:1 or 5:2?

🌟 Unlock Your Math Potential

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