Compare Circle Areas: Finding Area Ratio of 4 cm vs 10 cm Radius Circles

Q: Do I need to calculate the actual areas with π?

No! Since both circles have π in their area formulas, the π values cancel out when you divide. Just work with $ \frac{100}{16} $ instead of $ \frac{100π}{16π} $.

Q: How do I convert 100/16 to a mixed number?

Divide: 100 ÷ 16 = 6 with remainder 4. So $ \frac{100}{16} = 6\frac{4}{16} = 6\frac{1}{4} $ after simplifying the fraction part.

Q: What if the circles had different units, like cm and mm?

You'd need to convert to the same units first! For example, 1 cm = 10 mm, so a 4 cm radius becomes 40 mm before calculating the ratio.

Q: Is there a shortcut for finding area ratios?

Yes! For circles, the area ratio equals the square of the radius ratio. Here: $ \left(\frac{10}{4}\right)^2 = \left(\frac{5}{2}\right)^2 = \frac{25}{4} = 6\frac{1}{4} $

Area Ratios with Square Relationships

There are two circles.

One circle has a radius of 4 cm, while the other circle has a radius of 10 cm.

How many times greater is the area of the second circle than the area of the first circle?

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 How many times larger is circle 2's area than circle 1's area?

00:03 Let's use the formula for calculating circle area

00:10 Let's substitute the radius value according to the given data and solve for the area

00:17 This is circle 1's area

00:21 Let's use the same formula to calculate circle 2's area

00:29 This is circle 2's area

00:36 Let's divide the areas to find the ratio between them

00:52 Let's cancel out pi

00:57 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

There are two circles.

One circle has a radius of 4 cm, while the other circle has a radius of 10 cm.

How many times greater is the area of the second circle than the area of the first circle?

Step-by-step solution

The area of a circle is calculated using the following formula:

where r represents the radius.

Using the formula, we calculate the areas of the circles:

Circle 1:

$π\cdot4² =$

$π16$

Circle 2:

$\pi\cdot10²=$

$π100$

To calculate how much larger one circle is than the other (in other words - what is the ratio between them)

All we need to do is divide one area by the other.

$\frac{100}{16}=$

$6\frac{1}{4}$

Therefore the answer is 6 and a quarter!

Final Answer

$6\frac{1}{4}$

Key Points to Remember

Essential concepts to master this topic

Formula: Circle area equals π times radius squared
Technique: Calculate ratio by dividing larger area by smaller area
Check: Verify $\frac{100π}{16π} = \frac{100}{16} = 6\frac{1}{4}$ ✓

Common Mistakes

Avoid these frequent errors

Adding or subtracting radii instead of squaring
Don't compare radii directly like 10 - 4 = 6! Area depends on radius squared, not radius itself. A 10cm radius circle isn't just 2.5 times bigger - it's 6.25 times bigger because area scales with the square of radius. Always use $A = πr^2$ and compare the full areas.

Practice Quiz

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What is the ratio between the orange and gray parts in the drawing?

FAQ

Everything you need to know about this question

Why is the answer 6¼ and not just 2½ times bigger?

Great question! While the radius is 2.5 times bigger (10 ÷ 4 = 2.5), area grows with the square of radius. So you need to calculate $2.5^2 = 6.25 = 6\frac{1}{4}$ !

Do I need to calculate the actual areas with π?

No! Since both circles have π in their area formulas, the π values cancel out when you divide. Just work with $\frac{100}{16}$ instead of $\frac{100π}{16π}$ .

How do I convert 100/16 to a mixed number?

Divide: 100 ÷ 16 = 6 with remainder 4. So $\frac{100}{16} = 6\frac{4}{16} = 6\frac{1}{4}$ after simplifying the fraction part.

What if the circles had different units, like cm and mm?

You'd need to convert to the same units first! For example, 1 cm = 10 mm, so a 4 cm radius becomes 40 mm before calculating the ratio.

Is there a shortcut for finding area ratios?

Yes! For circles, the area ratio equals the square of the radius ratio. Here: $\left(\frac{10}{4}\right)^2 = \left(\frac{5}{2}\right)^2 = \frac{25}{4} = 6\frac{1}{4}$

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Compare Circle Areas: Finding Area Ratio of 4 cm vs 10 cm Radius Circles

Area Ratios with Square 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

Why is the answer 6¼ and not just 2½ times bigger?

Do I need to calculate the actual areas with π?

How do I convert 100/16 to a mixed number?

What if the circles had different units, like cm and mm?

Is there a shortcut for finding area ratios?

🌟 Unlock Your Math Potential

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