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Radius
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06.02 • Circle Geometry

Radius

Track the segment that connects the center to the boundary and use it as the basic length that sizes the whole circle.

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◯ Radius
Interactive diagram

Radius Diagram

Drag the endpoint on the circle and watch the radius keep one endpoint at the center and the other on the circumference.

Use the movable diagram to see what defines radius, how the labels relate to the figure, and what stays true as the board changes.

Definition: A radius is a segment from the center of a circle to a point on the circle.
Detailed definition

Understanding Radius

A radius is a segment from the center of a circle to a point on the circle. The same word is also used for the length of that segment, which is why students need to read carefully whether a problem is naming the segment or its measure.

All radii of the same circle have equal length. That simple fact explains why the circle stays perfectly round and why the diameter is always twice the radius.

On this page the radius is shown as a live segment, not just as a number in a formula. That makes it easier to connect the definition to circumference, area, and tangent geometry.

Key facts

Important ideas to remember

  • A radius is a segment from the center of a circle to a point on the circle.
  • In one circle, every radius has the same length.
  • A diameter is made from two radii placed end to end through the center.
  • A tangent is perpendicular to the radius drawn to the point of tangency.
Where it is used

Where radius shows up

  • Use radius when finding circumference, area, sector area, or arc length.
  • Use it in compass constructions because the chosen radius sets the size of the circle.
  • Use it in circle theorems where a radius meets a tangent or helps form a central angle.
Common mistakes

What to watch out for

  • Do not confuse radius with diameter; the radius reaches from center to edge, not from edge to edge.
  • Do not call any interior segment a radius unless one endpoint is the center and the other lies on the circle.
  • Do not switch length units when moving from the radius to formulas for circumference or area.
Worked examples

Radius examples

Use these worked examples to see the idea in a clean diagram first, then in the kind of reasoning students usually need for classwork, homework, or test practice.

Example 1

Example 1: Locating radius on one circle

Name the center and boundary points first so the diagram makes the term unmistakable.

  • Identify the center.
  • Find the exact segment, point, or measurement being named.
  • Check that the label matches the definition.

Result: The vocabulary is anchored to the right part of the circle.

Example 2

Example 2: Using radius before a calculation

Treat the diagram term as the reason a certain formula or fact becomes relevant.

  • Name the circle part clearly.
  • Choose the rule that belongs to that part.
  • Use the figure to confirm that the setup is correct.

Result: The calculation starts from the geometry instead of from a guessed formula.

For

Why this page helps

This page helps because radius is the measurement that holds the circle together. Students use it in formulas, in constructions, and in theorems about tangents, diameters, arcs, and sectors, so it needs to feel precise.

Do

What you can do here

  • Resize the circle by moving one radius endpoint and watch the measurement update immediately.
  • Compare radius with diameter and see their two-to-one relationship on the same figure.
  • Save a clear diagram that shows exactly where the radius begins and ends.
Learning outcome

What this page helps you do

These takeaways are meant to help you recognize the idea faster, read diagrams more accurately, and use the topic with more confidence in real problems.

1

Radius

Use radius confidently as the key length in circle formulas and constructions.

2

Radius

Separate the geometric segment from the numerical measure when a problem uses the same word in both ways.

3

Radius

Recognise how many other circle facts are built directly from the radius.

06

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Previous: Center

The center is the point equally distant from every point on the circle.

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Next: Diameter

A diameter is a chord passing through the center and equals two radii.

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