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?

February 17, 2026

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

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.

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

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?

Because while smaller circle area is:\[A_{\ ext{small}} = \pi \ imes 5^2 = 25\pi\]Then \(225\pi \div 25\pi = 9\), confirming the area increased by a factor of 9, proportional to \(3^2\).

Key Takeaways

Circumference relates to radius via \(C = 2\pi r\).- Radius scaling directly affects area through \(A = \pi r^2\), making it sensitive to squared changes.- Using proportional reasoning helps verify calculations efficiently.

Understanding these relationships makes solving circle-related questions fast, accurate, and intuitive.

Curious? Try This TodayUse the formula \(A = \pi r^2\) with \(r = 15\). Try calculating it step-by-step on paper—then check your answer with calculators or apps that show formulas in action. Geometry becomes not just a subject, but a hands-on puzzle!

Tagline: Master circle math: radius to area, scale with confidence.Keywords: circle area formula, radius and area relationship, circumference to radius, geometry problems explained, how to calculate circle area, scaling radius, π circle calculations

Need more geometry insights? Explore our guides on circle theorems, flying savant-style with clear formulas every time.