The radius is the distance from the center of a circle to its perimeter. It is a fundamental measurement that defines the size and shape of a circular object.
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The radius of a circle is half the diameter, and the diameter is twice the radius.
The formula for the circumference of a circle is $2\pi r$, where $r$ is the radius.
The formula for the area of a circle is $\pi r^2$, where $r$ is the radius.
The radius is a key measurement in trigonometric functions, such as sine, cosine, and tangent, which are used to describe the properties of a unit circle.
The radius is also an important concept in polar coordinates, where it represents the distance from the origin to a point on the plane.
Review Questions
Explain how the radius is related to the diameter and circumference of a circle.
The radius of a circle is half the diameter, and the diameter is twice the radius. The circumference of a circle is directly proportional to the radius, as shown by the formula $2\pi r$, where $r$ is the radius. This relationship between the radius, diameter, and circumference is fundamental to understanding the properties and measurements of a circle.
Describe the role of the radius in the unit circle and trigonometric functions.
In the context of the unit circle, the radius is always equal to 1. This means that the $x$-coordinate of a point on the unit circle is given by the cosine of the angle, and the $y$-coordinate is given by the sine of the angle. The radius is a key component in the definitions of the trigonometric functions, as it allows for the calculation of the sine, cosine, and tangent of an angle in the unit circle.
Explain how the radius is used in the formula for the area of a circle and discuss the significance of this relationship.
The formula for the area of a circle is $\pi r^2$, where $r$ is the radius. This formula demonstrates the direct relationship between the radius and the area of a circle. The radius is a fundamental measurement that determines the size and scale of a circular object, and understanding this relationship is crucial for calculating the area of circles in various applications, such as in geometry, engineering, and everyday life.
Related terms
Diameter: The diameter of a circle is the length of a straight line segment that passes through the center and connects two points on the circumference.
Circumference: The circumference of a circle is the distance around the outside of the circle, or the total distance covered by traveling once around the circle.
Arc Length: The arc length is the distance along the curved part of a circle between two points on the circumference.