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The area of a circle of radius 1 is units squared. Use scaling to explain why the area of a circle of radius ris ² units squared. •

The area of a circle of radius 1 is units squared. Use scaling to explain why the area of a circle of radius ris ² units squared. •

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**Concept Explained: Area of a Circle** The area of a circle with a radius of 1 is \(\pi\) square units. To understand why the area of a circle with a radius \(r\) is \(\pi \cdot r^2\) square units, consider the concept of scaling. ### Explanation 1. **Circle with Radius 1:** - The area is defined as \(\pi\) square units. 2. **Scaling the Circle:** - When you scale the circle's radius from 1 to \(r\), every linear dimension (like the radius) is multiplied by \(r\). - The area, being two-dimensional, is scaled by the square of this factor, which is \(r^2\). 3. **Conclusion:** - Therefore, the area of a circle with radius \(r\) is scaled by \(r^2\), resulting in an area of \(\pi \cdot r^2\) square units. Understanding this scaling principle helps visualize why the formula for the area of a circle scales with the square of the radius.
Transcribed Image Text:**Concept Explained: Area of a Circle** The area of a circle with a radius of 1 is \(\pi\) square units. To understand why the area of a circle with a radius \(r\) is \(\pi \cdot r^2\) square units, consider the concept of scaling. ### Explanation 1. **Circle with Radius 1:** - The area is defined as \(\pi\) square units. 2. **Scaling the Circle:** - When you scale the circle's radius from 1 to \(r\), every linear dimension (like the radius) is multiplied by \(r\). - The area, being two-dimensional, is scaled by the square of this factor, which is \(r^2\). 3. **Conclusion:** - Therefore, the area of a circle with radius \(r\) is scaled by \(r^2\), resulting in an area of \(\pi \cdot r^2\) square units. Understanding this scaling principle helps visualize why the formula for the area of a circle scales with the square of the radius.
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