Calculate the Area: Four x-cm Squares at y-cm Square Vertices

Q: Why is the answer 4x² + y² and not just x² + y²?

Because there are four separate squares with side length x, one at each vertex! Each has area x², so the total area from all four corner squares is $ 4 \times x^2 = 4x^2 $.

Q: Do the corner squares overlap with the central square?

No overlap! The corner squares are positioned at the vertices (corners) of the central square. They extend outward from the central square, creating additional area.

Q: How do I remember the area formula for squares?

Think: side × side = side². For a square with side length s, the area is always $ s^2 $. So x-cm squares have area $ x^2 $ and y-cm squares have area $ y^2 $.

Q: What if the corner squares were inside the central square instead?

Then you'd subtract the overlapping area! But in this problem, the diagram clearly shows the corner squares extending outward, so we add all areas together.

Q: Can I solve this by finding the perimeter first?

No! Perimeter measures the distance around the outside edge. Area measures the space inside the shape. These are completely different measurements that can't be converted into each other.

Area Calculation with Composite Geometric Shapes

At the vertices of a square with sides measuring y cm, 4 squares are drawn with lengths of x cm.

What is the area of the shape?

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find the area of the entire shape

00:03 In a square all sides are equal

00:07 We'll use the formula for calculating the area of a square (side times side)

00:12 This is the area of the small square

00:16 Let's write the formula for calculating the area of the entire shape

00:20 4 times the area of the small square plus the area of the large square

00:27 Again we'll use the formula for calculating the area of a square (side times side)

00:33 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

At the vertices of a square with sides measuring y cm, 4 squares are drawn with lengths of x cm.

What is the area of the shape?

Step-by-step solution

We will refer to two separate areas: the area of the square with side y and the total area of the four squares with sides x,

We'll use the formula for the area of a square with side b:

$S=b^2$ and therefore when applying it to the problem, we get that the area of the square with side y in the drawing is:

$S_1=y^2$ Next, we'll calculate the area of the square with side x in the drawing:

$S_2=x^2$ and to get the total area of the four squares in the drawing, we'll multiply this area by 4:

$4S_2=4x^2$ Therefore, the area of the required figure in the problem, which includes the area of the square with side y and the area of the four squares with side x is:

$S_1+4S_2=y^2+4x^2$ Therefore, the correct answer is A.

Final Answer

$4x^2+y^2$

Key Points to Remember

Essential concepts to master this topic

Rule: Total area equals sum of all individual shape areas
Technique: Calculate y² for central square plus 4x² for corner squares
Check: Verify units match and all shapes counted: 1 + 4 = 5 shapes ✓

Common Mistakes

Avoid these frequent errors

Counting overlapping areas or missing shapes
Don't add areas where shapes overlap or forget to count all four corner squares = wrong total area! The squares are separate with no overlap. Always identify each distinct shape: 1 central square (y²) + 4 corner squares (4x²) = y² + 4x².

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why is the answer 4x² + y² and not just x² + y²?

Because there are four separate squares with side length x, one at each vertex! Each has area x², so the total area from all four corner squares is $4 \times x^2 = 4x^2$ .

Do the corner squares overlap with the central square?

No overlap! The corner squares are positioned at the vertices (corners) of the central square. They extend outward from the central square, creating additional area.

How do I remember the area formula for squares?

Think: side × side = side². For a square with side length s, the area is always $s^2$ . So x-cm squares have area $x^2$ and y-cm squares have area $y^2$ .

What if the corner squares were inside the central square instead?

Then you'd subtract the overlapping area! But in this problem, the diagram clearly shows the corner squares extending outward, so we add all areas together.

Can I solve this by finding the perimeter first?

No! Perimeter measures the distance around the outside edge. Area measures the space inside the shape. These are completely different measurements that can't be converted into each other.

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Calculate the Area: Four x-cm Squares at y-cm Square Vertices

Area Calculation with Composite Geometric Shapes

❤️ 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

Why is the answer 4x² + y² and not just x² + y²?

Do the corner squares overlap with the central square?

How do I remember the area formula for squares?

What if the corner squares were inside the central square instead?

Can I solve this by finding the perimeter first?

🌟 Unlock Your Math Potential

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