Circle Area Calculation: Finding the Area with 1/2 cm Radius

Q: Why do I need to square the radius when it's already a fraction?

The circle area formula always requires squaring the radius, whether it's a whole number or fraction. $ A = \pi r^2 $ means radius squared, so $ \left(\frac{1}{2}\right)^2 = \frac{1}{4} $.

Q: How do I square a fraction like 1/2?

Square both the numerator and denominator separately: $ \left(\frac{1}{2}\right)^2 = \frac{1^2}{2^2} = \frac{1}{4} $. It's that simple!

Q: Should I convert π/4 to a decimal?

Keep it as a fraction! $ \frac{\pi}{4} $ is the exact answer. Converting to decimals introduces rounding errors and is usually not required unless specifically asked.

Q: What if the radius was 1/3 cm instead?

Same process! Square the radius: $ \left(\frac{1}{3}\right)^2 = \frac{1}{9} $, then multiply by π to get $ \frac{\pi}{9} $ cm².

Q: Why is the area so small when the radius is 1/2 cm?

Because we're squaring a fraction less than 1! When you square $ \frac{1}{2} $, you get $ \frac{1}{4} $, which makes the area even smaller. This makes sense - it's a very tiny circle!

Circle Area with Fractional Radius

What is the area of a round cake that has a radius of $\frac{1}{2}$ cm?

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find the area of the circle

00:04 Use the formula to calculate the circle's area

00:08 Substitute the appropriate values according to the given data

00:21 Write the power as multiplication

00:28 Be careful to multiply numerator by numerator and denominator by denominator

00:41 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

What is the area of a round cake that has a radius of $\frac{1}{2}$ cm?

Step-by-step solution

To solve this problem, we'll follow these steps:

Step 1: Identify the given information
Step 2: Apply the appropriate formula
Step 3: Perform the necessary calculations

Now, let's work through each step:
Step 1: The problem gives us that the radius $r = \frac{1}{2}$ cm.
Step 2: We'll use the formula for the area of a circle: $A = \pi r^2$ .
Step 3: Plugging in the radius, we have $A = \pi \left(\frac{1}{2}\right)^2$ . Calculating the square, we get $\left(\frac{1}{2}\right)^2 = \frac{1}{4}$ . Thus, the area becomes $A = \pi \times \frac{1}{4} = \frac{\pi}{4}$ .

Therefore, the solution to the problem is $\frac{\pi}{4}$ .

Final Answer

$\frac{\pi}{4}$

Key Points to Remember

Essential concepts to master this topic

Formula: Area of circle equals π times radius squared
Technique: Square the fraction: $\left(\frac{1}{2}\right)^2 = \frac{1}{4}$
Check: Area = $\pi \times \frac{1}{4} = \frac{\pi}{4}$ cm² ✓

Common Mistakes

Avoid these frequent errors

Forgetting to square the fractional radius
Don't just multiply π by 1/2 = π/2! This skips the crucial squaring step in the formula A = πr². Always remember to square the radius first: (1/2)² = 1/4, then multiply by π.

Practice Quiz

Test your knowledge with interactive questions

$\frac{2}{3}\times\frac{5}{7}=$

FAQ

Everything you need to know about this question

Why do I need to square the radius when it's already a fraction?

The circle area formula always requires squaring the radius, whether it's a whole number or fraction. $A = \pi r^2$ means radius squared, so $\left(\frac{1}{2}\right)^2 = \frac{1}{4}$ .

How do I square a fraction like 1/2?

Square both the numerator and denominator separately: $\left(\frac{1}{2}\right)^2 = \frac{1^2}{2^2} = \frac{1}{4}$ . It's that simple!

Should I convert π/4 to a decimal?

Keep it as a fraction! $\frac{\pi}{4}$ is the exact answer. Converting to decimals introduces rounding errors and is usually not required unless specifically asked.

What if the radius was 1/3 cm instead?

Same process! Square the radius: $\left(\frac{1}{3}\right)^2 = \frac{1}{9}$ , then multiply by π to get $\frac{\pi}{9}$ cm².

Why is the area so small when the radius is 1/2 cm?

Because we're squaring a fraction less than 1! When you square $\frac{1}{2}$ , you get $\frac{1}{4}$ , which makes the area even smaller. This makes sense - it's a very tiny circle!

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