inradius
C2 (Proficient User)Technical/Academic
Definition
Meaning
The radius of a circle that is tangent to all sides of a polygon, typically a triangle.
In geometry, the radius of the largest circle that can be inscribed within a given polygon, touching each side. For a triangle, it is the distance from the triangle's incenter to any of its sides.
Linguistics
Semantic Notes
A precise geometric term with no metaphorical use. Used almost exclusively in mathematics, engineering, and computational geometry.
Dialectal Variation
British vs American Usage
Differences
No significant differences in meaning or usage. Spelling and pronunciation are identical.
Connotations
Purely technical; carries no cultural or regional connotations.
Frequency
Extremely rare in general discourse. Frequency is identical in both academic/technical contexts.
Vocabulary
Collocations
Grammar
Valency Patterns
the inradius of [geometric figure]an inradius equal to [value]Vocabulary
Synonyms
Strong
Neutral
Weak
Vocabulary
Antonyms
Usage
Context Usage
Business
Virtually never used.
Academic
Used in geometry textbooks, research papers, and advanced mathematics courses.
Everyday
Never used in everyday conversation.
Technical
Used in engineering design, CAD software, computational geometry, and architectural planning.
Examples
By CEFR Level
- To find the area of the triangle, you can use the formula involving the semiperimeter and the inradius.
- The engineer calculated the inradius of the triangular component to ensure the drill hole would be centrally located and not breach any edges.
- In planar geometry, an equilateral triangle has a specific ratio between its side length and its inradius.
Learning
Memory Aids
Mnemonic
Think 'IN-side RADIUS' — the radius of the circle that fits INside the shape.
Conceptual Metaphor
N/A (Highly technical term without metaphorical application)
Watch out
Common Pitfalls
Translation Traps (for Russian speakers)
- Direct equivalent is 'радиус вписанной окружности'. No false friend trap exists due to the term's specificity.
Common Mistakes
- Confusing 'inradius' with 'circumradius' (the radius of the circle that passes through all vertices).
- Using 'inradius' for 3D shapes (for a sphere inscribed in a polyhedron, the term is 'insphere radius' or 'inradius' with explicit 3D context).
Practice
Quiz
What is the relationship between the inradius (r), area (A), and semiperimeter (s) of a triangle?
FAQ
Frequently Asked Questions
Yes, 'inradius' can be used for any polygon that has an incircle (a circle tangent to all sides), such as regular polygons. For a triangle, an incircle always exists.
For a regular polygon, the apothem is the distance from the center to the midpoint of a side, which is identical to the inradius. For irregular polygons or triangles, 'inradius' is the correct general term.
Conceptually, if a polygon degenerates (e.g., a very flat triangle), the inradius approaches zero. A circle cannot be inscribed in a shape with no interior angle bisector intersection, so the inradius is undefined in such cases.
It is used in mechanical engineering to determine the maximum size of a shaft or hole that fits within a component, in architecture for planning interior circular elements, and in computer graphics for mesh generation and collision detection algorithms.