MathematicsChapter 9: Geometry and Polygons
Polygons and Angle Sums
A polygon is a closed plane figure made of line segments.
Learning objectives
- Explain polygons and angle sums in clear mathematical language.
- Use the main rule, formula, test, or construction correctly.
- Work through a simple example and show each important step.
- Identify common mistakes and write an exam-ready conclusion.
What this grid covers
A polygon is a closed plane figure made of line segments. The least number of sides is 3 (a triangle). An n-gon is a polygon with n sides. A regular polygon has all sides equal and all interior angles equal. A square is regular. A non-square rectangle is irregular (angles equal, sides not all equal). A rhombus that is not a square is irregular (sides equal, angles not all equal). Pentagon means 5 sides; hexagon means 6.
Where / symbols
- closed plane figure
- A polygon is a closed plane figure made of line segments.
- least number of sides is 3
- The least number of sides is 3 (a triangle).
- n-gon
- An n-gon is a polygon with n sides.
- regular
- A regular polygon has all sides equal and all interior angles equal.
- irregular
- A non-square rectangle is irregular (angles equal, sides not all equal).
WHAT — Core idea
A polygon is a closed plane figure made of line segments. The least number of sides is 3 (a triangle). An n-gon is a polygon with n sides. A regular polygon has all sides equal and all interior angles equal. A square is regular. A non-square rectangle is irregular (angles equal, sides not all equal). A rhombus that is not a square is irregular (sides equal, angles not all equal). Pentagon means 5 sides; hexagon means 6.
HOW — Rules, method, and examples
Interior angle sum of a convex n-gon is (n-2)180 deg. Each extra side adds one more triangle: triangle: (3-2)180 = 180 deg; 1803 = 60 deg. quadrilateral: (4-2)180 = 360 deg; 3604 = 90 deg (square). pentagon: (5-2)180 = 540 deg; 5405 = 108 deg. hexagon: (6-2)180 = 720 deg; 7206 = 120 deg. heptagon: (7-2)180 = 900 deg; 9007 deg (not a whole number).
WHY — Meaning and connections
An interior angle sits between two adjacent sides. An exterior angle sits between a side and the extension of the next side. An interior angle and its adjacent exterior angle add to 180 deg (they form a straight line). For a convex polygon the exterior angles, one at each vertex, sum to 360 deg. A regular interior angle is [(n-2)*180]/n deg.
HOW — Answer in an exam
Least polygon is 3 sides. Regular needs equal sides and equal interior angles: a square yes; a non-square rectangle no; a non-square rhombus no. Memorise pentagon 540, hexagon 720, heptagon 900. Convex exterior sum is 360 deg, not (n-2)*180.
Interior angle sum of a convex n-gon is (n-2)*180 deg. Each extra side adds one more triangle:
| n-gon | Interior sum (n-2)*180 | Regular interior angle |
|---|---|---|
| 3. triangle | (3-2)*180 = 180 deg | 180/3 = 60 deg |
| 4. quadrilateral | (4-2)*180 = 360 deg | 360/4 = 90 deg (square) |
| 5. pentagon | (5-2)*180 = 540 deg | 540/5 = 108 deg |
| 6. hexagon | (6-2)*180 = 720 deg | 720/6 = 120 deg |
| 7. heptagon | (7-2)*180 = 900 deg | 900/7 deg (not a whole number) |
Explanation
An interior angle sits between two adjacent sides. An exterior angle sits between a side and the extension of the next side. An interior angle and its adjacent exterior angle add to 180 deg (they form a straight line). For a convex polygon the exterior angles, one at each vertex, sum to 360 deg. A regular interior angle is [(n-2)*180]/n deg.
In simple words
Least polygon is 3 sides. Regular needs equal sides and equal interior angles: a square yes; a non-square rectangle no; a non-square rhombus no. Memorise pentagon 540, hexagon 720, heptagon 900. Convex exterior sum is 360 deg, not (n-2)*180.
Key points
- closed plane figure
- least number of sides is 3
- n-gon
- regular
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Polygons and Angle Sums
Sticky Notes · Quick Revision
WHAT
- A polygon is a closed plane figure made of line segments.
Key Points
- A polygon is a closed plane figure made of line segments.
- Interior angle sum of a convex n-gon is (n-2)*180 deg.
- triangle: (3-2)*180 = 180 deg.
- quadrilateral: (4-2)*180 = 360 deg.
HOW
- Interior angle sum of a convex n-gon is (n-2)*180 deg.
Examples
- triangle: (3-2)*180 = 180 deg
- quadrilateral: (4-2)*180 = 360 deg
- pentagon: (5-2)*180 = 540 deg
- hexagon: (6-2)*180 = 720 deg
Common Mistakes
- Correction: Least polygon is 3 sides.
- Correction: Regular needs equal sides and equal interior angles: a square yes; a non-square rectangle no; a non-square rhombus no.
- Correction: Memorise pentagon 540, hexagon 720, heptagon 900.
Exam Tip
- Least polygon is 3 sides. Regular needs equal sides and equal interior angles: a square yes; a non-square rectangle no; a non-square rhombus no. Memorise pentagon 540, hexagon 720, heptagon 900. Convex exterior sum is 360 deg, not (n-2)*180.
60-second revision
- A polygon is a closed plane figure made of line segments.
- Interior angle sum of a convex n-gon is (n-2)*180 deg.
- triangle: (3-2)*180 = 180 deg.
- quadrilateral: (4-2)*180 = 360 deg.
- An interior angle sits between two adjacent sides.
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