Calculate Circle Area When Rectangle Area = 28X: Geometric Relationship Problem

Q: How do I know the circle's diameter equals the rectangle's width?

From the diagram, you can see the circle touches both vertical sides of the rectangle. This means the circle's diameter exactly equals the rectangle's width.

Q: Why do I divide the rectangle area by its height first?

Because Area = length × width, so if you know the area (28x) and height (7), you can find the width: $ width = \frac{28x}{7} = 4x $.

Q: How does rectangle width become circle diameter?

The circle is inscribed in the rectangle, meaning it fits perfectly inside. The circle's diameter must equal the rectangle's width to touch both sides.

Q: What's the difference between radius and diameter here?

The diameter is the full width (4x), while the radius is half that distance (2x). Always use radius in the area formula: $ A = \pi r^2 $.

Circle Areas with Rectangle Dimension Relationships

The area of the rectangle in the drawing is 28X cm².

What is the area of the circle?

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Express the circumference of the circle using X

00:03 The short side is a diameter according to the given data

00:07 Apply the formula to calculate the area of the rectangle: side(7) multiplied by side(2R)

00:10 Insert the relevant values into the formula and solve for R

00:19 Isolate R

00:25 This is the expression for radius(R) using X

00:29 Apply the formula to calculate the circle's circumference

00:36 Substitute the relevant values into the formula and solve to determine the circumference expression

00:45 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

The area of the rectangle in the drawing is 28X cm².

What is the area of the circle?

Step-by-step solution

Let's draw the center of the circle and we can divide the diameter of the circle into two equal radii

Now let's calculate the length of the radii as follows:

$7\times2r=28x$

$14r=28x$

We'll divide both sides by 14:

$r=\frac{28}{14}x$

$r=2x$

Let's calculate the circumference of the circle:

$P=2\pi\times r=2\pi\times2x=4\pi x$

Final Answer

$4\pi x$

Key Points to Remember

Essential concepts to master this topic

Identify Dimensions: Circle diameter equals rectangle width from given measurements
Calculate Radius: Divide area by height: 28x ÷ 7 = 4x, then r = 2x
Verify Formula: Area = $\pi r^2 = \pi(2x)^2 = 4\pi x$ ✓

Common Mistakes

Avoid these frequent errors

Using circumference formula instead of area formula
Don't calculate P = 2πr = 4πx and think this is the area! Circumference measures the circle's boundary, not its interior space. Always use A = πr² for circle area.

Practice Quiz

Test your knowledge with interactive questions

Look at the rectangle below.

Side DC has a length of 1.5 cm and side AD has a length of 9.5 cm.

What is the perimeter of the rectangle?

FAQ

Everything you need to know about this question

How do I know the circle's diameter equals the rectangle's width?

From the diagram, you can see the circle touches both vertical sides of the rectangle. This means the circle's diameter exactly equals the rectangle's width.

Why do I divide the rectangle area by its height first?

Because Area = length × width, so if you know the area (28x) and height (7), you can find the width: $width = \frac{28x}{7} = 4x$ .

How does rectangle width become circle diameter?

The circle is inscribed in the rectangle, meaning it fits perfectly inside. The circle's diameter must equal the rectangle's width to touch both sides.

What's the difference between radius and diameter here?

The diameter is the full width (4x), while the radius is half that distance (2x). Always use radius in the area formula: $A = \pi r^2$ .

Why isn't the answer just 28x like the rectangle?

Different shapes have different areas even with the same dimensions! A rectangle uses all its space, but a circle has curved edges that don't fill the corners completely.

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Calculate Circle Area When Rectangle Area = 28X: Geometric Relationship Problem

Circle Areas with Rectangle Dimension Relationships

❤️ 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 the circle's diameter equals the rectangle's width?

Why do I divide the rectangle area by its height first?

How does rectangle width become circle diameter?

What's the difference between radius and diameter here?

Why isn't the answer just 28x like the rectangle?

🌟 Unlock Your Math Potential

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