Polygons and Quadrilaterals

Learning Intentions

• determine if a polygon is convex or non-convex
• determine if a polygon is regular or irregular
• classify a polygon by the number of sides it has
• To know what a quadrilateral is
• classify a quadrilateral as a parallelogram, rectangle, rhombus, square, trapezium or kite based on a diagram or a description

Pre-requisite Summary

• Understand that a polygon is a closed 2D shape made only from straight line segments
• Know that sides meet at vertices and that polygons have no curved edges
• Be able to count the number of sides of a shape accurately
• Understand that equal side and angle markings in diagrams give information about shape properties
• Know that a quadrilateral is a polygon with $4$ sides
• Understand that parallel sides remain the same distance apart and never meet
• Understand that right angles are ${90}^{\circ }$
• Be able to distinguish between “all sides equal”, “opposite sides equal”, and “one pair of equal adjacent sides”

Worked Examples

Worked Example 1

a) Explain what makes a polygon convex.
b) Explain what makes a polygon non-convex.
c) Decide whether a given polygon with an inward “dent” is convex or non-convex.

Worked Example 2

a) Explain what makes a polygon regular.
b) Explain what makes a polygon irregular.
c) Decide whether a polygon with all sides and all angles equal is regular or irregular.

Worked Example 3

Classify each polygon by number of sides:
a) a polygon with $3$ sides
b) a polygon with $5$ sides
c) a polygon with $8$ sides

Worked Example 4

a) State what a quadrilateral is.
b) Decide whether a $4$-sided polygon is a quadrilateral.
c) Explain why a triangle is not a quadrilateral.

Worked Example 5

Classify each quadrilateral from its properties:
a) opposite sides parallel and equal, but no right-angle information
b) $4$ right angles and opposite sides equal
c) $4$ equal sides and $4$ right angles

Worked Example 6

Classify each quadrilateral from its description:
a) exactly one pair of parallel sides
b) two pairs of equal adjacent sides
c) opposite sides parallel and all sides equal, but no right angles marked

Problems

Problem 1

a) Explain what makes a polygon convex.
b) Explain what makes a polygon non-convex.
c) Decide whether a given polygon with an inward “dent” is convex or non-convex.

Problem 2

a) Explain what makes a polygon regular.
b) Explain what makes a polygon irregular.
c) Decide whether a polygon with unequal side lengths is regular or irregular.

Problem 3

Classify each polygon by number of sides:
a) a polygon with $4$ sides
b) a polygon with $6$ sides
c) a polygon with $10$ sides

Problem 4

a) State what a quadrilateral is.
b) Decide whether a $4$-sided polygon is a quadrilateral.
c) Explain why a pentagon is not a quadrilateral.

Problem 5

Classify each quadrilateral from its properties:
a) opposite sides parallel and equal, but no right-angle information
b) $4$ right angles and opposite sides equal
c) $4$ equal sides and $4$ right angles

Problem 6

Classify each quadrilateral from its description:
a) exactly one pair of parallel sides
b) two pairs of equal adjacent sides
c) opposite sides parallel and all sides equal, but no right angles marked

Exercises

Understanding and Fluency

1. Decide whether each polygon is convex or non-convex:
   a) a pentagon with all vertices pointing outward
   b) a hexagon with one interior “dent”
   c) a quadrilateral with no interior “dent”

2. Decide whether each polygon is regular or irregular:
   a) a hexagon with all sides and angles equal
   b) a pentagon with unequal angles
   c) a square

3. Classify each polygon by number of sides:
   a) $3$ sides
   b) $4$ sides
   c) $5$ sides

4. Classify each polygon by number of sides:
   a) $6$ sides
   b) $7$ sides
   c) $8$ sides

5. State whether each shape is a quadrilateral:
   a) a shape with $4$ straight sides
   b) a shape with $3$ straight sides
   c) a shape with $5$ straight sides

6. Classify each quadrilateral from the description:
   a) opposite sides parallel and equal
   b) $4$ right angles
   c) $4$ equal sides and $4$ right angles

7. Classify each quadrilateral from the description:
   a) exactly one pair of parallel sides
   b) two pairs of equal adjacent sides
   c) all sides equal, but no right angles are shown

8. State all possible names that apply:
   a) a quadrilateral with $4$ right angles and all sides equal
   b) a quadrilateral with $4$ equal sides
   c) a quadrilateral with opposite sides parallel and equal

9. Match each name to its property:
   a) parallelogram
   b) trapezium
   c) kite

10. Mixed practice:
    a) Is a square regular or irregular?
    b) Is a concave polygon convex or non-convex?
    c) How many sides does a quadrilateral have?

Reasoning

11. Explain why a polygon with an inward “dent” is non-convex.

12. A student says a rectangle is not a parallelogram. Explain the mistake.

13. Explain why a square can also be described as a rectangle and a rhombus.

14. A student says any $4$-sided shape with equal sides must be a square. Explain why this is incorrect.

15. Explain why a regular polygon must have both equal sides and equal angles.

16. A student says a trapezium has two pairs of parallel sides. Explain why this is incorrect under the usual school definition.

Problem-solving

17. A shape has $4$ sides. Its opposite sides are parallel, and each angle is ${90}^{\circ }$. Classify the shape as precisely as possible.

18. A quadrilateral has side lengths $5,5,7,7$ in adjacent pairs, with no parallel sides marked. Classify the quadrilateral.

19. A polygon has $8$ sides, all equal, and all interior angles equal. Classify it by number of sides and by regularity.

20. A floor tile is a quadrilateral with all sides equal and opposite sides parallel, but its angles are not right angles. Classify the tile.

21. A sign has exactly one pair of parallel sides. Classify the quadrilateral.

22. A diagram shows a $4$-sided shape with all sides equal and all angles equal. State every correct classification that applies.

Potential Misunderstandings

• Students may think any closed shape is a polygon, even if it has curved edges
• Students may confuse convex and non-convex by looking only at side lengths
• Students may think regular means only “symmetrical-looking” rather than requiring all sides and all angles equal
• Students may confuse the number of vertices with the number of sides when classifying polygons
• Students may forget that a quadrilateral must have exactly $4$ sides
• Students may think a square belongs to only one category, rather than also fitting the definitions of rectangle, rhombus, and parallelogram
• Students may think all quadrilaterals with equal sides are squares, ignoring angle information
• Students may confuse a kite with a rhombus because both can have equal sides, but the equality pattern is different
• Students may confuse a trapezium with a parallelogram by miscounting the number of parallel side pairs