Circle Circumference Function: Express in Terms of Radius 'a'

Q: Why is it 2πa and not just πa?

The circumference formula is C = 2πr, where the 2 comes from the relationship between diameter and radius. Since diameter = 2r, and circumference = π × diameter, we get C = π(2r) = 2πr.

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

Circumference measures the distance around the circle (like a fence), while area measures the space inside the circle (like a lawn). Circumference uses 2πr, area uses πr².

Q: Do I need to calculate the numerical value of 2πa?

No! Since 'a' is a variable, you leave your answer as $ 2\pi a $. This is the function form that works for any value of radius 'a'.

Q: Why does the answer look like y = 2πa instead of C = 2πa?

The question asks for a function, so we use 'y' to represent the output (circumference) and 'a' as the input (radius). Both forms mean the same thing!

Q: How can I remember this isn't the area formula?

Think about units! Circumference measures length (like inches), so it's linear. Area measures space (like square inches), so it has the variable squared. No squares in circumference!

Circle Formula with Variable Radius

A circle has a radius of a.

Choose the function that expresses its circumference.

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Express the circumference of the circle

00:03 We will use the formula for calculating the circumference of a circle

00:07 We will substitute appropriate values according to the given data, and solve to find the circumference

00:11 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 of a.

Choose the function that expresses its circumference.

Step-by-step solution

The goal of this problem is to find the function that represents the circumference of the circle when the radius is $a$ .

The formula for the circumference of a circle is:

$C = 2\pi r$

We are given that the radius $r$ is $a$ .

Using the formula $C = 2\pi r$ , substitute $a$ for $r$ :

$C = 2\pi a$

Thus, the function that expresses the circumference of the circle is:

$y = 2\pi a$

Among the given choices, $y = 2\pi a$ corresponds to choice number 4.

Final Answer

$y=2\pi a$

Key Points to Remember

Essential concepts to master this topic

Formula: Circumference of a circle is always C = 2πr
Technique: Substitute the given radius: C = 2π(a) = 2πa
Check: Units match and π appears once (not squared) ✓

Common Mistakes

Avoid these frequent errors

Confusing circumference and area formulas
Don't use πa² for circumference = wrong formula entirely! That's the area formula, which gives square units instead of linear units. Always use C = 2πr for circumference, which gives the distance around the circle.

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 2πa and not just πa?

The circumference formula is C = 2πr, where the 2 comes from the relationship between diameter and radius. Since diameter = 2r, and circumference = π × diameter, we get C = π(2r) = 2πr.

What's the difference between circumference and area?

Circumference measures the distance around the circle (like a fence), while area measures the space inside the circle (like a lawn). Circumference uses 2πr, area uses πr².

Do I need to calculate the numerical value of 2πa?

No! Since 'a' is a variable, you leave your answer as $2\pi a$ . This is the function form that works for any value of radius 'a'.

Why does the answer look like y = 2πa instead of C = 2πa?

The question asks for a function, so we use 'y' to represent the output (circumference) and 'a' as the input (radius). Both forms mean the same thing!

How can I remember this isn't the area formula?

Think about units! Circumference measures length (like inches), so it's linear. Area measures space (like square inches), so it has the variable squared. No squares in circumference!

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