Question: A circle has a circumference of 31.4 meters. What is the area of the circle?

Understanding the area of a circle starts with knowing its circumference—a fundamental relationship in geometry. If you’ve ever wondered how to calculate the area given just the circle’s circumference, this article will guide you step-by-step through the process using a real-world example: a circle with a circumference of 31.4 meters.

What Is Circumference and Why Does It Matter?

Understanding the Context

Circumference is the total distance around the edge of a circle, measured in meters (or any unit of length). The formula to calculate circumference is:

$$
C = 2\pi r
$$

where:
- $ C $ = circumference
- $ \pi $ (pi) ≈ 3.14
- $ r $ = radius of the circle

Since the circumference is 31.4 meters, we can solve for the radius.

Circumference And Area Of A Circle Worksheet

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Circumference And Area Of A Circle Worksheet
Circumference And Area Of A Circle Worksheet
Area And Circumference Of A Circle Worksheet - Adriansonfifth
Circumference of a Circle Worksheets - Math Monks - Worksheets Library
How to Find the Circumference and Area of a Circle: 5 Steps

Key Insights

Step 1: Solve for the Radius

Using the circumference formula:

$$
31.4 = 2\pi r
$$

To isolate $ r $, divide both sides by $ 2\pi $:

$$
r = rac{31.4}{2\pi}
$$

Continue Reading

But problem states sum is 210, so must be consistent.
10+12+14+16+18+20+22+28=148+28=176; too big.
But arithmetic sequences must have integer n.

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Final Thoughts

Substituting $ \pi pprox 3.14 $:

$$
r = rac{31.4}{2 \ imes 3.14} = rac{31.4}{6.28} = 5 \ ext{ meters}
$$

So, the radius of the circle is 5 meters.

Step 2: Use the Radius to Calculate the Area

The area $ A $ of a circle is calculated with the formula:

$$
A = \pi r^2
$$

Now plug in $ r = 5 $:

$$
A = \pi \ imes 5^2 = \pi \ imes 25 pprox 3.14 \ imes 25 = 78.5 \ ext{ square meters}
$$

Final Answer

A circle with a circumference of 31.4 meters has an area of 78.5 square meters.