Circle Area Function: Selecting the Correct Formula for Radius a

Q: Why is it πa² and not just a²?

The π (pi) is essential because circles are curved! The value π ≈ 3.14 accounts for the circular shape. Without π, you'd just have the area of a square with side length a.

Q: What's the difference between area and circumference formulas?

Area measures the space inside: $ A = \pi r^2 $Circumference measures the distance around: $ C = 2\pi r $Notice area has r² while circumference has just r!

Q: Can I write the answer as A = πa² instead of y = πa²?

Both are correct! A = πa² uses A for area, while y = πa² treats it as a function. Since the question asks for a function, y = πa² is the preferred format.

Q: Why do some answers have 2π or 4 in them?

Those are wrong formulas! $ 2\pi a $ is circumference (distance around), $ 2a $ is diameter, and $ 4a $ isn't a circle measurement at all.

Q: How do I remember which formula to use?

Think about what you're measuring: Area = space inside = squared units = $ \pi r^2 $Circumference = distance around = linear units = $ 2\pi r $

Area Formulas with Variable Radius

A circle has a radius a.

Choose the function that expresses the area of the circle.

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Express the area of the circle

00:04 We will use the formula for calculating the area of a circle

00:09 We will substitute the appropriate values according to the given data, and solve to find the area

00:14 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

A circle has a radius a.

Choose the function that expresses the area of the circle.

Step-by-step solution

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

Identify the formula needed to compute the area of a circle.
Substitute the given radius into the formula.
Simplify the expression to match with the provided choices.

Now, let's work through the solution:

Step 1: The problem gives us the radius of the circle as $a$ .

Step 2: We'll apply the formula for the area of a circle:
$A = \pi r^2$

Step 3: Substitute $r = a$ :
$A = \pi a^2$

Therefore, the function expressing the area of the circle is $\boxed{y = \pi a^2}$ .

Final Answer

$y=\pi a^2$

Key Points to Remember

Essential concepts to master this topic

Formula: Area of circle equals π times radius squared
Technique: Substitute r = a into A = πr² to get A = πa²
Check: Verify units are squared (length²) and π coefficient is present ✓

Common Mistakes

Avoid these frequent errors

Using circumference formula instead of area formula
Don't use C = 2πr when finding area = wrong formula entirely! The circumference formula gives perimeter, not the space inside. Always use A = πr² for area calculations.

Practice Quiz

Test your knowledge with interactive questions

Complete:

The missing value of the function point:

$$ f(x)=x^2 $$

$$ f(?)=16 $$

FAQ

Everything you need to know about this question

Why is it πa² and not just a²?

The π (pi) is essential because circles are curved! The value π ≈ 3.14 accounts for the circular shape. Without π, you'd just have the area of a square with side length a.

What's the difference between area and circumference formulas?

Area measures the space inside: $A = \pi r^2$
Circumference measures the distance around: $C = 2\pi r$
Notice area has r² while circumference has just r!

Can I write the answer as A = πa² instead of y = πa²?

Both are correct! A = πa² uses A for area, while y = πa² treats it as a function. Since the question asks for a function, y = πa² is the preferred format.

Why do some answers have 2π or 4 in them?

Those are wrong formulas! $2\pi a$ is circumference (distance around), $2a$ is diameter, and $4a$ isn't a circle measurement at all.

How do I remember which formula to use?

Think about what you're measuring: Area = space inside = squared units = $\pi r^2$
Circumference = distance around = linear units = $2\pi r$

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