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

00:00 Find the ratio of areas between the circles

00:03 We'll use the formula for calculating circle area

00:09 We'll substitute the radius and solve to find the area

00:15 The circle's radius equals half the diameter

00:24 Circles with equal radii also have equal areas

00:34 And this is the solution to the question

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

Find the area of the circle according to the drawing.

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

🌟 Unlock Your Math Potential

Get unlimited access to all 18 Circle for Ninth Grade questions, detailed video solutions, and personalized progress tracking.

📹

Unlimited Video Solutions

Step-by-step explanations for every problem

📊

Progress Analytics

Track your mastery across all topics

🚫

Ad-Free Learning

Focus on math without distractions

No credit card required • Cancel anytime

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

Circle Area Comparison with Radius-Diameter Conversion

❤️ 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 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

More Questions

Area of a Circle for ninth grade

Calculate Circle Area Ratio: Comparing 4 cm vs 10 cm Diameter Circles

Area of a Geometric Flower: Solving with Variables 2x, 1.2x, and 1.5x

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

Find Circle Area: Perpendicular Radii with Distance y+2

Ratio

Circle Radius Comparison: 14 cm Red vs 7 cm Blue Diameter

Circle Radius Comparison: Finding the Ratio of 24cm vs 12cm Diameters

Deltoid Diagonal Problem: Finding Secondary Diagonal Length with 7:1 Ratio

Deltoid Diagonal Problem: Finding Secondary Diagonal Length Using 5:1 Ratio