Question

Use the formula for arc length to show that the circumference of the circle \(x^{2}+y^{2}=1\) is \(2 \pi\).

Short answer

The circumference of the circle is \(2\pi\).

Step-by-step solution

Identify the Circle's Equation

The equation given is \(x^2 + y^2 = 1\). This represents a circle centered at the origin \((0,0)\) with a radius \(r = 1\).

Recall the Formula for Arc Length

The arc length \(L\) of a circle is given by the formula \(L = r \theta\), where \(\theta\) is the central angle in radians and \(r\) is the radius. For a full circle, \(\theta = 2\pi\).

Apply Arc Length Formula to Find the Circumference

Substitute the values into the arc length formula. Since \(\theta = 2\pi\) and \(r = 1\), the circumference \(L\) is \(L = 1 \times 2\pi = 2\pi\).

Conclusion

Thus, using the arc length formula, the circumference of the circle defined by \(x^{2}+y^{2}=1\) is confirmed to be \(2\pi\).

Key concepts

Arc Length

The concept of arc length is crucial when dealing with circles and their circumferences. The arc length refers to the distance you measure along a section of the curve of the circle. To calculate this, we use the equation:

• \( L = r \theta \)

Here, \(L\) is the arc length, \(r\) is the radius of the circle, and \(\theta\) is the angle measured in radians that subtends the arc at the circle's center.

The formula essentially allows you to determine how much of the circle's circumference an arc takes up, depending on the angle. For a full circle, the angle is \(2\pi\) radians, which makes the arc length equal to the circle's circumference. This straightforward relationship helps in understanding that the circumference of a circle with radius \(r = 1\) is \(2\pi\), a fundamental property derived from the circle equation itself.

Circle Equation

Understanding circle equations is vital in geometry and algebra. The standard equation of a circle is:

• \( (x - h)^2 + (y - k)^2 = r^2 \)

Where \((h, k)\) are the coordinates of the center of the circle, and \(r\) is the radius.

In the given exercise, the equation \(x^2 + y^2 = 1\) is a specific instance of this equation where the circle is centered at the origin, \((0,0)\), and has a radius of \(1\). This simplified form makes many calculations more straightforward and intuitive.

Such equations not only help in sketching circles on a coordinate plane but also in analytic geometry tasks, like verifying the properties of circles, which include understanding and calculating arc lengths and circumferences.

Radians

Radians are a unit of angular measure used in mathematics, especially when dealing with circles and trigonometric functions. Unlike degrees, radians provide a direct relationship between the radius of a circle and the arc length.

• A full circle is \(2\pi\) radians.
• One radian is the angle created when the arc length is equal to the radius.

Using radians simplifies many mathematical formulas. For example, the arc length formula \(L = r \theta\) directly incorporates radians, streamlining calculations when determining parts of a circle.

Radians connect easily with the circumference of a circle because when \(\theta = 2\pi\), the circle is complete, mimicking nature's cycles and symmetries in a neat numeric manner. This ties together the understanding of circle measures and angles within the realm of geometry and trigonometry.