Calculate Circle Radius Given Circumference C=81: Geometric Problem

Q: Why do we divide by 2π instead of just π?

Because the circumference formula is C = 2πr, not C = πr. The factor 2 comes from the relationship between radius and diameter. To isolate r, you must divide by the complete coefficient 2π.

Q: Should I use 3.14 or a more precise value for π?

For accurate answers, use π ≈ 3.14159 or your calculator's π button. Using just 3.14 can give slightly different results, especially for larger circumferences.

Q: What if I need an exact answer instead of a decimal?

The exact answer is $ r = \frac{81}{2\pi} $. Only convert to decimal when specifically asked for a numerical approximation.

Q: Can I work backwards to check my answer?

Absolutely! Calculate $ 2\pi \times 12.891 $ and see if you get close to 81. This backward check confirms your radius is correct.

Circle Radius with Circumference Formula

Calculate the radius based on the circumference given in the figure:

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find the circle's radius (R)

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

00:11 We'll substitute appropriate values according to the given data and calculate to find the radius

00:23 We'll isolate radius R

00:35 We'll substitute the value of pi and calculate to find radius R

00:48 And this is the solution to the problem

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Calculate the radius based on the circumference given in the figure:

Step-by-step solution

To solve the problem of finding the radius based on the given circumference, we follow these steps:

Step 1: Identify the given information: Circumference $C = 81$ .
Step 2: Use the formula for circumference: $C = 2\pi r$ .
Step 3: Solve the equation for the radius $r$ .

Let's execute each step in detail:

Step 1: We have $C = 81$ .

Step 2: The formula relating circumference and radius is:

\ C = 2\pi r \

Step 3: Substitute the given circumference into the formula:

\ 81 = 2\pi r \

Step 4: Solve for the radius $r$ :

Dividing both sides by $2\pi$ :

\ r = \frac{81}{2\pi} \

Using the numerical approximation for $\pi$ , i.e., $\pi \approx 3.14159$ :

\ r \approx \frac{81}{2 \times 3.14159} \

\ r \approx \frac{81}{6.28318} \

\ r \approx 12.891 \

Therefore, the radius of the circle is approximately $12.891$ .

Final Answer

12.891

Key Points to Remember

Essential concepts to master this topic

Formula: Use C = 2πr to find radius from circumference
Technique: Solve r = C/(2π) = 81/(2π) ≈ 12.891
Check: Verify 2π(12.891) ≈ 81 matches given circumference ✓

Common Mistakes

Avoid these frequent errors

Confusing radius and diameter formulas
Don't use C = πd formula and divide by π only = wrong answer of 25.8! This gives the diameter, not radius. Always use C = 2πr and divide by 2π to get the correct radius.

Practice Quiz

Test your knowledge with interactive questions

$$ r=2 $$

Calculate the circumference.

FAQ

Everything you need to know about this question

Why do we divide by 2π instead of just π?

Because the circumference formula is C = 2πr, not C = πr. The factor 2 comes from the relationship between radius and diameter. To isolate r, you must divide by the complete coefficient 2π.

Should I use 3.14 or a more precise value for π?

For accurate answers, use π ≈ 3.14159 or your calculator's π button. Using just 3.14 can give slightly different results, especially for larger circumferences.

How do I know if 12.891 is reasonable for this circle?

Check if it makes sense: a radius of about 13 gives a circumference of roughly 2 × 3.14 × 13 ≈ 82, which is very close to our given C = 81!

What if I need an exact answer instead of a decimal?

The exact answer is $r = \frac{81}{2\pi}$ . Only convert to decimal when specifically asked for a numerical approximation.

Can I work backwards to check my answer?

Absolutely! Calculate $2\pi \times 12.891$ and see if you get close to 81. This backward check confirms your radius is correct.

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