Circle Radius Calculation: Understanding the Equation x2 y2 144

Have you ever wondered how to find the radius of a circle given its equation? In this article, we will explore the concept using the specific equation x2 y2 144. This is a fundamental topic in geometry and is crucial for understanding more complex mathematical concepts. By the end of this article, you will have a clear understanding of how to determine the radius of a circle using basic algebra and the Pythagorean theorem.

Understanding the Circle Equation: x2 y2 144

Let's start with a straightforward equation of a circle, x2 y2 144. This equation is often written in the standard form of a circle's equation, which is (x - h)2 (y - k)2 r2. In this equation, (h, k) are the coordinates of the circle's center, and r is the radius.

The Center of the Circle

In the given equation, there is neither an x nor a y term added or subtracted. This indicates that the center of the circle is at the origin, or the point (0, 0) on the coordinate plane. This simplifies our calculation significantly, as we don't need to account for any translations.

The Radius of the Circle

The key to finding the radius is in the constant term on the right side of the equation, which is 144. In the standard form, 144 represents the square of the radius, r2. Therefore, to find r, we take the square root of 144:

r √144 12

This means the radius of the circle is 12 units. This result can also be interpreted using the Pythagorean theorem, which states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. In our circle, any point (x, y) on the circumference of the circle can be seen as the endpoints of a right-angled triangle, with the radius as the hypotenuse.

Applying the Pythagorean Theorem

To see this more clearly, let's apply the Pythagorean theorem to the equation:

r2 x2 y2

Given that we already know x2 y2 144 from the circle's equation, it follows that:

r √144 12

Thus, any point (x, y) on the circle's circumference is such that the relationship between the radius and the coordinates continues to hold true.

Conclusion

In conclusion, the radius of the circle x2 y2 144 is 12 units. This understanding is not only crucial for solving geometric problems but also forms the basis for more advanced topics in mathematics. By mastering these fundamentals, you can tackle a wide range of mathematical challenges with confidence.

Key Takeaways

The circle equation x2 y2 144 tells us that the center is at the origin and the radius is 12 units. Using the Pythagorean theorem, we can confirm that the radius of the circle is 12 units. Understanding these concepts is essential for solving more complex mathematical problems.

Resources

For further reading and practice, consider the following resources:

Math is Fun - Circle Wikipedia - Circle Khan Academy - Circles

By exploring these resources, you can deepen your understanding of circles and related geometric concepts.