Circle Area Calculation: Can a 5-Unit Chord Determine the Complete Circle?

Q: Why can't I use the chord length to find the area?

A chord is just a straight line segment inside the circle. The same 5-unit chord could belong to a small circle (where it's almost a diameter) or a huge circle (where it's tiny compared to the diameter). You need the radius or diameter to find area.

Q: What information would I need to calculate the area?

You need either: The radius (Area = $ \pi r^2 $)The diameter (radius = diameter ÷ 2)The chord length plus the distance from chord to center

Q: Could there be multiple circles with the same chord?

Yes, infinitely many! Think of it this way: you can draw countless circles of different sizes that all contain the same 5-unit line segment. That's why chord length alone isn't enough.

Q: What if the chord was a diameter?

If the problem stated that AB was a diameter (not just a chord), then you could find the area! Diameter = 5 means radius = 2.5, so Area = $ \pi(2.5)^2 = 6.25\pi $.

Circle Properties with Insufficient Information

Look at the circle in the diagram.

AB is a chord.

Is it possible to calculate 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 If possible, calculate the area of the circle

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

00:07 We don't know anything about the radius, therefore it cannot be calculated

00:12 The given chord doesn't pass through the circle's center, therefore it's not a diameter

00:15 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

Look at the circle in the diagram.

AB is a chord.

Is it possible to calculate the area of the circle?

Step-by-step solution

Since AB is just a chord and we know nothing else about the diameter or the radius, we cannot calculate the area of the circle.

Final Answer

It is not possible.

Key Points to Remember

Essential concepts to master this topic

Rule: Circle area requires radius or diameter, not just chord length
Technique: Multiple circles can contain the same chord: chord = 5 doesn't determine radius
Check: Ask yourself: do I have radius, diameter, or center information? ✓

Common Mistakes

Avoid these frequent errors

Assuming chord length determines circle area
Don't think that knowing chord AB = 5 means you can find the area! Infinite circles of different sizes can contain the same 5-unit chord. Always check if you have radius, diameter, or enough geometric constraints to determine the circle uniquely.

Practice Quiz

Test your knowledge with interactive questions

The center of the circle in the diagram is O.

What is the area of the circle?

FAQ

Everything you need to know about this question

Why can't I use the chord length to find the area?

A chord is just a straight line segment inside the circle. The same 5-unit chord could belong to a small circle (where it's almost a diameter) or a huge circle (where it's tiny compared to the diameter). You need the radius or diameter to find area.

What information would I need to calculate the area?

You need either:

The radius (Area = $\pi r^2$ )
The diameter (radius = diameter ÷ 2)
The chord length plus the distance from chord to center

Could there be multiple circles with the same chord?

Yes, infinitely many! Think of it this way: you can draw countless circles of different sizes that all contain the same 5-unit line segment. That's why chord length alone isn't enough.

Is this a trick question?

Not a trick - it's testing your understanding of what information is sufficient vs insufficient. In geometry, always identify what you know and what you need before attempting calculations.

What if the chord was a diameter?

If the problem stated that AB was a diameter (not just a chord), then you could find the area! Diameter = 5 means radius = 2.5, so Area = $\pi(2.5)^2 = 6.25\pi$ .

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