Yes, the radius is indeed half of the diameter. This fundamental relationship is a core concept in the geometry of circles.
Understanding Radius and Diameter
To fully grasp this relationship, it's important to define both terms precisely:
- Radius (r): As per the reference, the radius is defined as the distance from the centre of a circle to any point on its boundary. Imagine a string tied from the very center of a circle to its edge; the length of that string is the radius.
- Diameter (d): While not explicitly defined in full detail in the provided reference for the diameter's definition, the reference states its relationship to the radius. Generally, the diameter is a straight line segment that passes through the center of a circle and has its endpoints on the circle's boundary. It's essentially the widest part of a circle.
The Relationship: 2r = d
The reference explicitly states: "The radius is half of the diameter; 2r=d." This means that if you know the radius of a circle, you can find its diameter by simply multiplying the radius by two. Conversely, if you know the diameter, you can find the radius by dividing the diameter by two.
This relationship can be expressed by the following formulas:
- d = 2r (Diameter equals two times the radius)
- r = d/2 (Radius equals half of the diameter)
Radius vs. Diameter: A Quick Comparison
| Feature | Radius (r) | Diameter (d) |
|---|---|---|
| Definition | Distance from the center to any point on the boundary | A line segment passing through the center, connecting two points on the boundary |
| Length | Half the length of the diameter | Twice the length of the radius |
| Formula | r = d/2 | d = 2r |
| Chord? | Not a chord | Every diameter is a chord (a line segment joining two points on the circle) |
Practical Insights and Examples
Understanding the relationship between radius and diameter is crucial for various calculations in geometry and real-world applications.
- Calculating Diameter from Radius:
- If a circle has a radius of 5 cm, its diameter would be:
- d = 2 * r
- d = 2 * 5 cm = 10 cm
- If a circle has a radius of 5 cm, its diameter would be:
- Calculating Radius from Diameter:
- If a circular table has a diameter of 120 cm, its radius would be:
- r = d / 2
- r = 120 cm / 2 = 60 cm
- If a circular table has a diameter of 120 cm, its radius would be:
Diameter as a Special Chord
The reference also highlights an important distinction regarding chords: "The line segment that joins two points on the circle is a chord. Every diameter is a chord, but not every chord is a diameter."
- A chord is any straight line segment whose endpoints both lie on the circle.
- A diameter is a specific type of chord that has the unique property of passing directly through the circle's center. Because it passes through the center, it is the longest possible chord in any given circle.
Understanding these concepts is foundational for further study in geometry, including calculating circumference, area, and volume of circular and spherical shapes.
