Problem 76 - Algebra 2

Question

What is the area of the circle whose equation is \((x+1)^{2}+(y+1)^{2}=1 ?\) Round your answer to the nearest hundredth.

Step-by-Step Solution

Answer

The area of the circle is \(\pi\) (approximately 3.14)

Step 1: Identify the radius

The equation of the circle is given as \((x+1)^{2}+(y+1)^{2}=1\). This equation can be written in the standard form \((x-h)^2+(y-k)^2=r^2\). Comparing these two equations, we find that the radius r is 1.

Step 2: Calculate the area

Substitute the radius into the area formula of a circle, which is \(Area=\pi r^2\). Substituting r=1, we get \(Area=\pi*(1)^2=\pi\).

Key Concepts

Circle Area

Understanding how to calculate the area of a circle is essential in geometry. The area is a measure of the total space contained within the circle's boundary. To find the area, we use the formula: - \[Area = \pi r^2\] Where - \(\pi\) is a constant (approximately 3.14159), representing the ratio of the circumference of any circle to its diameter. - \(r\) is the radius of the circle. In practical problems, once the radius is known, finding the area simply involves plugging the value of the radius into the formula. In our example, the circle equation \(x+1)^2+(y+1)^2=1\) indicated a radius \(r\) of 1. Thus, the area is: \[Area = \pi \times 1^2 = \pi\] This means the area of the circle is \(\pi\) square units, or approximately 3.14 square units when rounded.

Standard Form of a Circle

The standard form of a circle's equation offers a simple way to identify key attributes of the circle, particularly its center and radius. The formula is typically written as: - \[(x-h)^2+(y-k)^2=r^2\] Where: - \((h, k)\) are the coordinates of the circle's center. - \(r\) is the radius. The given equation \(x+1)^2+(y+1)^2=1\) reveals that the circle's center is at \((-1, -1)\) and the radius is 1 because it equates to the standard form by reorganizing the elements. By understanding the standard form, we quickly deduce these characteristics, simplifying the process of locating the circle on a graph and computing further values, like the area.

Radius Calculation

The radius of a circle is crucial because it defines the circle's size and is used in various formulas, such as for area or circumference. In the context of a circle's equation, the radius is represented by \(r\) in the standard form: - \[(x-h)^2 + (y-k)^2 = r^2\] To find the radius from an equation like \(x+1)^2+(y+1)^2=1\), you compare it with the standard form \(x-h)^2+(y-k)^2=r^2\). The values of \((h, k)\) correspond to the center, and \(r^2\) is the squared radius. For our equation, since \(r^2 = 1\), we take the square root to find \(r\), which results in \(r = 1\). The radius is straightforward to calculate from the equation, providing vital information for further calculations of the circle's properties like its area and circumference.