A radius is the distance from the center of a circle to any point on its circumference. Let me show you this with a visual example. Here we have a circle with its center marked in red. The radius is the straight line segment that connects the center to any point on the edge of the circle. This blue line represents one radius. An important property to remember is that all radii in a circle are equal in length, no matter which point on the circumference you choose.
Now let's explore the key properties of radius. First, every circle has infinite radii, and all of them are exactly equal in length. Watch as I draw multiple radius lines from the center to different points on the circumference. Notice how each one has the same length, which we call 'r'. An important relationship to remember is that the diameter equals two times the radius. The diameter is the longest distance across the circle, passing through the center. Finally, if we rotate a radius around the center, it traces out the entire circle, demonstrating how the radius defines the circle's size and shape.
Now let's explore how radius appears in important mathematical formulas. The three key formulas involving radius are: circumference equals 2 pi r, area equals pi r squared, and diameter equals 2 r. Let me demonstrate with a specific example. If the radius is 3 units, then the circumference is 2 pi times 3, which equals 6 pi. The area is pi times 3 squared, which equals 9 pi. And the diameter is 2 times 3, which equals 6. Watch how changing the radius affects all these values. As the radius increases, both the circumference and area grow, but notice that the area grows much faster because it depends on r squared.
Radius appears everywhere in our daily lives. In a pizza, the radius is the distance from the center to the edge, which determines the size of each slice. On a clock, the hands act as radii, extending from the center to show the time. In a bicycle wheel, the spokes are radii connecting the hub to the rim. In a circular garden, the radius determines how far plants are from the center. Let's solve a practical problem: if a circular track has a circumference of 31.4 centimeters, what is its radius? Using the formula r equals C divided by 2 pi, we get r equals 31.4 divided by 2 pi, which equals 5 centimeters.
Let's clarify the differences between radius and other circle elements to avoid confusion. The radius goes from the center to any point on the edge. The diameter goes from edge to edge, passing through the center, and is always twice the radius length. A chord also connects two points on the edge, but it doesn't have to pass through the center. An arc is the curved path along the circle's edge, unlike the straight radius. Key properties to remember: radius equals half the diameter, radius always passes through the center, and radius is straight while an arc is curved. In our example with radius 2, the diameter is 4, this particular chord is about 3.46 units, and this arc segment is about 3.14 units long.