Table of Contents
Understanding Radius and Diameter
In this chapter we look closely at two basic measurements in a circle: the radius and the diameter. We assume you already know what a circle is from the parent chapter.
A circle is completely determined by its center and its radius. The diameter is closely related to the radius, and both are used in almost every calculation involving circles.
Definitions
A circle has a special point called its center. Distances from this center to points on the circle define:
- Radius
A radius of a circle is a line segment from the center of the circle to any point on the circle.
The length of this segment is also called the radius (often written as $r$). - Diameter
A diameter of a circle is a line segment that: - passes through the center of the circle, and
- has both endpoints on the circle.
The length of any such segment is called the diameter (often written as $d$).
In words:
- Radius: center to circle.
- Diameter: across the circle, going through the center.
Every circle has infinitely many radii (you can draw a radius in any direction) and infinitely many diameters, but all radii have the same length, and all diameters have the same length.
Relationship Between Radius and Diameter
The diameter is made of two radii placed end to end in a straight line through the center. So the diameter is exactly twice the radius.
The basic formulas are:
$$
d = 2r
$$
and equivalently,
$$
r = \frac{d}{2}
$$
These are the main relationships you will use:
- If you know the radius, double it to get the diameter.
- If you know the diameter, divide by $2$ to get the radius.
Working with Units
Radius and diameter are lengths, so they use standard length units like:
- centimeters (cm), meters (m), millimeters (mm),
- inches (in), feet (ft), etc.
They always use the same units. If the radius is $5\text{ cm}$, the diameter is $10\text{ cm}$, not $10\text{ m}$ or another unit.
Examples:
- If $r = 7\text{ mm}$, then $d = 2 \times 7\text{ mm} = 14\text{ mm}$.
- If $d = 12\text{ cm}$, then $r = 12\text{ cm} \div 2 = 6\text{ cm}$.
Using Symbols and Notation
Common notation for a circle with center $O$ and radius $r$:
- The circle itself might be named $\odot O$ (circle with center $O$).
- A radius might be written as segment $OA$, where $O$ is the center and $A$ is a point on the circle.
- A diameter might be written as segment $AB$, where both $A$ and $B$ lie on the circle and the center $O$ lies on segment $AB$.
The lengths are then written as:
- $OA = r$ (radius length),
- $AB = d$ (diameter length).
Visual Identification
When you see a drawing of a circle:
- To recognize a radius:
- Look for a segment starting at the center and ending on the circle.
- It does not need to be horizontal or vertical; any direction is fine.
- To recognize a diameter:
- Look for a segment with both endpoints on the circle that passes through the center.
- It should “cut” the circle into two equal halves.
Not every line across the circle is a diameter. A chord that does not pass through the center is not a diameter, even if it looks long. For a segment to be a diameter, the center must lie on it.
Comparing Lengths Inside a Circle
- Every radius has the same length, because every point on the circle is the same distance from the center.
- Every diameter has the same length, because a diameter is always two radii in a straight line.
From $d = 2r$ we also know:
- Any diameter is longer than any radius of the same circle (twice as long, in fact).
Simple Numeric Examples
- Radius known:
- A circle has radius $r = 4\text{ cm}$.
Then the diameter is:
$$
d = 2r = 2 \times 4\text{ cm} = 8\text{ cm}.
$$ - Diameter known:
- A circle has diameter $d = 15\text{ m}$.
Then the radius is:
$$
r = \frac{d}{2} = \frac{15\text{ m}}{2} = 7.5\text{ m}.
$$ - Word description:
- “The distance from one side of the circular plate to the opposite side, passing through the center, is $20\text{ cm}$.”
This is a diameter, so $d = 20\text{ cm}$, and
$$
r = \frac{20\text{ cm}}{2} = 10\text{ cm}.
$$
Common Misconceptions
- Confusing radius and diameter:
- Radius: from center to edge.
- Diameter: all the way across through the center.
- Thinking there is only one radius:
- There are infinitely many, but they all have equal length.
- Thinking any line across is a diameter:
- It must pass through the center to be a diameter.
Remember: if you can clearly see the center point lying on the segment, it is a diameter; otherwise it is just a chord, not a diameter.
Why Radius and Diameter Matter
Many important circle formulas (covered in other chapters) are written in terms of $r$ or $d$. You will often need to:
- Convert between radius and diameter using $d = 2r$ and $r = d/2$.
- Decide whether a given measurement is a radius (center to edge) or a diameter (across the whole circle).
A strong grasp of radius and diameter makes all later work with circles—such as working with arcs and sectors, and using area or circumference formulas—much more straightforward.