Introduction to Circle Area Calculation and Radius Increase

Understanding the mathematical principles behind the change in the surface area of a circle when its radius increases is crucial for many applications in geometry and real-world scenarios. This article delves into the calculation of the increase in area when the radius of a circle is increased by 25%.

Initial Concepts

The surface area of a circle (which is the same as the area) is given by the formula A πr2, where r represents the radius of the circle. We will explore how this formula applies when the radius increases by 25%.

Calculation of the Increase in Area

Method 1

Let the original radius of a circle be r. When the radius increases by 25%, the new radius becomes 125% of r, which can be written as 1.25r. The new area, using the area formula for a circle, is given by:

Area π(1.25r)2

Simplifying this expression:

Area π(1.5625r2)

Therefore, the new area is 1.5625 times the original area, which means the area has increased by 56.25%.

Method 2

Another way to look at the problem is by expressing the original and new areas in terms of the original radius:

Original Area πr2

When the radius increases by 25%, the new radius is 5/4 of r. Therefore:

New Area π(5r/4)2

Simplifying this expression:

New Area π(25r2/16)

The increased area can be found by subtracting the original area from the new area and then dividing by the original area:

Increased Area π(25r2/16) - πr2 πr2(25/16 - 1) πr2(9/16)

Expressing the percentage increase:

Percentage Increase (9/16) * 100 56.25%

Conclusion

In conclusion, when the radius of a circle is increased by 25%, the area of the circle increases by 56.25%. This is a fundamental concept in geometry that has practical applications in various fields, including physics, engineering, and architecture.

Relevance to SEO

Keywords that are frequently searched for include circle area calculation, radius increase, and circle surface area. Understanding these concepts and presenting a clear, concise explanation of how and why the area increases by 25% can help improve the relevance and engagement of a webpage on search engines like Google.