Question**: A circle has a radius that is 3 times the radius of another circle with a circumference of 10π meters. What is the area of the larger circle? - Decision Point
How to Calculate the Area of a Larger Circle Using Radius Relationships – A Step-by-Step Explanation
How to Calculate the Area of a Larger Circle Using Radius Relationships – A Step-by-Step Explanation
When faced with a geometric problem involving circles, one of the most common challenges is understanding how changes in radius affect area and circumference. In this article, we’ll solve a classic question: A circle has a radius that is 3 times the radius of another circle with a circumference of 10π meters. What is the area of the larger circle?
The Problem: Circle Radius and Area Relationship
Understanding the Context
Let’s break the problem into clear steps using precise formulas and logical reasoning.
Step 1: Find the radius of the smaller circle
We start with the given data: the circumference of the smaller circle is 10π meters.
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Key Insights
Recall the formula for the circumference of a circle:
\[
C = 2\pi r
\]
Given \( C = 10\pi \), solve for \( r \):
\[
2\pi r = 10\pi
\]
Divide both sides by \( \pi \):
\[
2r = 10
\]
\[
r = 5 \ ext{ meters}
\]
So, the radius of the smaller circle is 5 meters.
Step 2: Determine the radius of the larger circle
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We’re told the radius of the larger circle is 3 times that of the smaller one:
\[
R = 3 \ imes 5 = 15 \ ext{ meters}
\]
Step 3: Calculate the area of the larger circle
Now that we know the radius of the larger circle (15 meters), use the area formula:
\[
A = \pi R^2
\]
\[
A = \pi (15)^2 = \pi \ imes 225 = 225\pi \ ext{ square meters}
\]
Final Answer: The Area is 225π square meters
This calculation demonstrates how proportional changes in radius directly impact area—a key concept in geometry and spatial reasoning. By recalling circumference and area formulas, and applying simple multiplication and substitution, this problem becomes clear and solvable.
Bonus Tip: Why Area Scales with the Square of Radius
Remember, while circumference grows linearly with radius (\(C \propto r\)), area grows quadratically (\(A \propto r^2\)). Since the larger circle has a radius 3 times bigger, its area is \(3^2 = 9\) times greater than the smaller one:
\[
9 \ imes (10\pi) = 90\pi \ ext{ square meters}
\]
But wait—our direct calculation gave \(225\pi\). Why the difference?
