Compare Circle Areas: Finding the Ratio When r₁ = 6cm and d₂ = 12cm

Q: Why do I need to convert diameter to radius?

The area formula $ A = \pi r^2 $ specifically uses radius, not diameter. Since radius = diameter ÷ 2, you must convert first. Using diameter directly gives you 4 times the actual area!

Q: How do I compare two circle areas?

Calculate each area separately using $ A = \pi r^2 $, then find the ratio by dividing: $ \frac{A_2}{A_1} $. If the ratio equals 1, the areas are equal.

Q: What if one circle has radius and another has diameter?

Convert everything to the same unit (either all radius or all diameter). In this problem, Circle 1 has r = 6 cm and Circle 2 has d = 12 cm, so r = 6 cm. Both radii are equal!

Q: Why are the areas equal when the numbers look different?

The key insight is that radius of 6 cm and diameter of 12 cm describe the same size circle! Since 12 ÷ 2 = 6, both circles have identical radius = 6 cm.

Q: How can I quickly tell if two circles are the same size?

If both given as radii: compare directlyIf both given as diameters: compare directlyIf mixed: convert to same unit first, then compare

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:12 Let's find the ratio of areas between these circles.

00:16 We'll use the formula: Area equals Pi times radius squared.

00:21 Next, substitute the radius to calculate each circle's area.

00:27 Remember, the radius is half of the diameter of the circle.

00:36 So, circles with the same radius have the same area.

00:46 And that's how we solve this problem.

Step-by-step written solution

Follow each step carefully to understand the complete solution

1

Understand the problem

There are two circles.

The length of the radius of circle 1 is 6 cm.

The length of the diameter of circle 2 is 12 cm.

How many times greater is the area of circle 2 than the area of circle 1?

2

Step-by-step solution

1. Restate the problem: We are given two circles. Circle 1 has a radius of 6 cm, and Circle 2 has a diameter of 12 cm. We need to determine how many times greater the area of Circle 2 is compared to Circle 1. 2. Key information: - Radius of Circle 1: 6 cm - Diameter of Circle 2: 12 cm, which implies the radius is half of the diameter, i.e., 6 cm. 3. Potential approach: Calculate the area of both circles and find the ratio of the area of Circle 2 to the area of Circle 1. 4. Key formula: The area of a circle is given by

A = \pi r^2

where

r

is the radius. 5. Appropriate approach: Use the formula for the area of a circle to find the areas of the two circles and compare them. 6. Steps: - Compute the area of Circle 1 using its radius. - Compute the area of Circle 2 using its radius. - Calculate the ratio of the areas of Circle 2 to Circle 1. 7. Assumptions: Circles are perfect geometrical circles, and

\pi

is a constant. 8. Break down: Since the radii of both circles are equal (both 6 cm), their areas will be identical. 9. Special conditions: None are expected beyond confirming equal areas due to equal radii. 10. Instructions: Ensure clarity and correctness in the solution. 11. Compare with choices: Verify if the areas are indeed equal as suggested by choice 4. 12. Common mistakes: Misunderstanding "diameter" and "radius" could lead to incorrect calculations. 13. Changing variables: Altering the radius or diameter affects circle areas proportionally to the square of the radius ratio.

To solve the problem, let's follow the necessary steps:

Step 1: Identify the given values for each circle.
Step 2: Use the formula for the area of a circle, $A = \pi r^2$ , to calculate both areas.
Step 3: Compare the areas to determine the ratio.

Now, let's go through each step:
Step 1: We know:

Circle 1 has a radius $r_1 = 6$ cm.
Circle 2 has a diameter of 12 cm, so its radius $r_2 = \frac{12}{2} = 6$ cm.

Step 2: Using the formula $A = \pi r^2$ , calculate the area of each circle:

For Circle 1: $A_1 = \pi (6)^2 = 36\pi$ square centimeters.
For Circle 2: $A_2 = \pi (6)^2 = 36\pi$ square centimeters.

Step 3: Compare the areas by calculating the ratio:
The ratio of the area of Circle 2 to Circle 1 is:

\frac{A_2}{A_1} = \frac{36\pi}{36\pi} = 1

This means that the areas of Circle 1 and Circle 2 are identical.

Therefore, the solution to the problem is that the areas are equal.

3

Final Answer

They are equal.

Key Points to Remember

Essential concepts to master this topic

Conversion Rule: Radius equals half the diameter for any circle
Technique: Use $A = \pi r^2$ where Circle 2 has radius = 12÷2 = 6 cm
Check: Both circles have radius 6 cm, so areas are $36\pi$ square cm ✓

Common Mistakes

Avoid these frequent errors

Using diameter directly in area formula
Don't substitute diameter = 12 into $A = \pi r^2$ = $\pi (12)^2 = 144\pi$ ! The area formula requires radius, not diameter. Always convert diameter to radius first by dividing by 2.

Practice Quiz

Test your knowledge with interactive questions

Calculate the area of a circle with a radius of 5 cm.

FAQ

Everything you need to know about this question

Why do I need to convert diameter to radius?

+

The area formula $A = \pi r^2$ specifically uses radius, not diameter. Since radius = diameter ÷ 2, you must convert first. Using diameter directly gives you 4 times the actual area!

How do I compare two circle areas?

+

Calculate each area separately using $A = \pi r^2$ , then find the ratio by dividing: $\frac{A_2}{A_1}$ . If the ratio equals 1, the areas are equal.

What if one circle has radius and another has diameter?

+

Convert everything to the same unit (either all radius or all diameter). In this problem, Circle 1 has r = 6 cm and Circle 2 has d = 12 cm, so r = 6 cm. Both radii are equal!

Why are the areas equal when the numbers look different?

+

The key insight is that radius of 6 cm and diameter of 12 cm describe the same size circle! Since 12 ÷ 2 = 6, both circles have identical radius = 6 cm.

How can I quickly tell if two circles are the same size?

+

If both given as radii: compare directly
If both given as diameters: compare directly
If mixed: convert to same unit first, then compare

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Compare Circle Areas: Finding the Ratio When r₁ = 6cm and d₂ = 12cm

Circle Area Comparison with Radius-Diameter Conversion

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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 do I need to convert diameter to radius?

How do I compare two circle areas?

What if one circle has radius and another has diameter?

Why are the areas equal when the numbers look different?

How can I quickly tell if two circles are the same size?

🌟 Unlock Your Math Potential

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