incircle
LowTechnical / Academic (Mathematics)
Definition
Meaning
A circle drawn inside a polygon, especially a triangle, that touches all of its sides.
In mathematics, the largest possible circle that fits entirely within a polygon, tangent to all of its sides. It also functions as a verb meaning to draw such a circle within a polygon.
Linguistics
Semantic Notes
Primarily a technical term in geometry. The verbal sense is less common but valid. The noun is sometimes hyphenated (in-circle) or written as two words (inscribed circle).
Dialectal Variation
British vs American Usage
Differences
No significant difference in meaning or usage, but spelling conventions may lead to 'centre' vs. 'center' in related definitions. The hyphenated form 'in-circle' is slightly more common in British technical writing.
Connotations
None beyond the technical mathematical context.
Frequency
Equally rare in both dialects, confined to mathematical discourse.
Vocabulary
Collocations
Grammar
Valency Patterns
incircle of [a polygon/triangle]to incircle [a polygon/triangle]Vocabulary
Synonyms
Strong
Neutral
Weak
Vocabulary
Antonyms
Phrases
Idioms & Phrases
- “None”
Usage
Context Usage
Business
Not used.
Academic
Used exclusively in geometry and related mathematical disciplines.
Everyday
Almost never used.
Technical
Standard term for the unique circle tangent to all sides of a triangle or tangential polygon.
Examples
By Part of Speech
verb
British English
- The next step is to incircle the triangle.
- Can you incircle this regular pentagon?
American English
- First, incircle the triangle to find the incenter.
- The software can automatically incircle any polygon.
adverb
British English
- Not applicable
American English
- Not applicable
adjective
British English
- The incircle centre is found at the intersection of the angle bisectors.
American English
- The incircle radius is a key measurement.
Examples
By CEFR Level
- A triangle can have a circle inside it. This circle is called an incircle.
- The incircle of a triangle touches all three sides.
- We learned how to draw an incircle in geometry class.
- The radius of the incircle is calculated using the formula r = A/s, where A is the area and s is the semiperimeter.
- To construct the incircle, you must first locate the incenter.
- In tangential polygons, the incircle is central to many area and perimeter theorems.
- The mathematician derived a new proof for the existence of the incircle in cyclic quadrilaterals.
Learning
Memory Aids
Mnemonic
Imagine a circle IN a shape, touching all its sides INSIDE—IN-CIRCLE.
Conceptual Metaphor
CONTAINMENT (a circle contained within boundaries, touching them perfectly).
Watch out
Common Pitfalls
Translation Traps (for Russian speakers)
- Do not confuse with 'окружность' (any circle). The precise Russian equivalent is 'вписанная окружность'. Avoid translating literally as 'внутренний круг', which is too vague.
Common Mistakes
- Confusing 'incircle' with 'circumcircle' (which goes around the polygon).
- Using it in non-geometric contexts.
- Misspelling as 'ensircle' or 'incercle'.
Practice
Quiz
What is the incircle of a triangle?
FAQ
Frequently Asked Questions
No, it is a specialised term used almost exclusively in geometry and mathematics.
Yes, though less common. It means 'to draw or construct an incircle within a polygon'.
An incircle is inside a polygon and touches its sides. A circumcircle is outside, passing through all its vertices.
No. Only tangential polygons (where all sides are tangent to a single circle) have an incircle. All triangles and some special quadrilaterals do.