The radius of a circle with a 10-foot diameter is exactly 5 feet.
Understanding the relationship between a circle's diameter and its radius is fundamental in geometry. The diameter is the distance across the circle, passing through its center. The radius is the distance from the center of the circle to any point on its edge.
The Relationship Between Radius and Diameter
The most crucial concept here is that the diameter is always twice the length of the radius. Conversely, the radius is always half the length of the diameter. This direct relationship makes calculating one from the other very straightforward.
- Formula:
- Diameter = 2 × Radius
- Radius = Diameter / 2
Calculating the Radius
Given a circle with a diameter of 10 feet, we can easily find its radius using the formula:
- Identify the Diameter: The problem states the circle has a 10-foot diameter.
- Apply the Formula: Radius = Diameter / 2
- Substitute and Solve: Radius = 10 feet / 2 = 5 feet
Therefore, a circle that measures 10 feet across its widest point (its diameter) will have a radius of 5 feet.
Visualizing the Dimensions
Imagine cutting a circle exactly in half through its center. The length of that cut is the diameter. Now, if you measure from the exact center of that circle to the edge, that's the radius. It makes sense that the radius would be half the length of the diameter.
Key Geometric Terms
To further clarify, here's a quick overview of related terms:
| Term | Definition | Relationship to Radius (r) |
|---|---|---|
| Radius | The distance from the center of a circle to any point on its circumference. | r |
| Diameter | The distance across a circle passing through its center. | 2r |
| Circumference | The total distance around the edge of a circle. | 2πr or πd |
| Area | The amount of space enclosed within the circle's boundary. | πr² |
Practical Applications
Knowing how to determine the radius from a given diameter is useful in various real-world scenarios:
- Construction and Design: When building circular structures, designing round tables, or specifying pipe sizes, engineers and designers frequently work with diameter and radius measurements. For instance, if you need to build a circular garden with a 10-foot diameter, you'll know to place a central pivot point and measure out 5 feet for the edge.
- Home Improvement: From selecting the right size for a circular rug to fitting a new light fixture, understanding these measurements ensures proper fit and aesthetic appeal.
- Crafts and Hobbies: Projects involving circular shapes, such as knitting, pottery, or model building, often require precise radius calculations.
By understanding the simple relationship between diameter and radius, you can easily calculate one from the other, providing a foundation for more complex geometric problems.
