043i - Classifying Polygons

Lesson 43 — Classifying Other Polygons

Strand: Space | Descriptor: AC9M7SP02 | Duration: 45 minutes

Learning Intentions

• To name and classify polygons by their number of sides.
• To distinguish regular from irregular, and convex from concave polygons.

Success Criteria

I can:

1. Name polygons with to sides.
2. Distinguish a regular polygon from an irregular one.
3. Distinguish a convex polygon from a concave one.
4. State the number of diagonals and lines of symmetry of a regular polygon.

Warmup

(6 minutes — naming recall, mini whiteboards)

Name the polygon with:

1. sides
2. sides
3. sides
4. sides
5. sides
6. sides

(Answers: triangle; pentagon; hexagon; octagon; quadrilateral; decagon.)

Prompt: Where else do you meet these prefixes? (Pentagon — the building; octagon — a stop sign; decade — ten years; hexagonal — honeycomb.)

Activities

Activity 1 — Explicit Instruction: Naming and Regularity (12 min)

Definition. A polygon is a closed two-dimensional shape with straight sides.

The naming table — build with the class:

Sides	Name	Sides	Name
	triangle		octagon
	quadrilateral		nonagon
	pentagon		decagon
	hexagon		hendecagon
	heptagon		dodecagon

Regular versus irregular.

• A regular polygon has all sides equal AND all angles equal.
• If either condition fails, it is irregular.

The critical point — both conditions are needed. Display two counterexamples:

• A rhombus has all sides equal but unequal angles — irregular.
• A rectangle has all angles equal but unequal sides — irregular.

Only the square satisfies both, making it the regular quadrilateral.

We do: Regular or irregular?

1. A square
2. An equilateral triangle
3. A rectangle cm by cm
4. A hexagon with all sides cm but two angles larger than the rest
5. A stop sign

(Answers: regular; regular; irregular; irregular; regular octagon.)

Activity 2 — Convex and Concave (10 min)

Definitions.

• A convex polygon has every interior angle less than . All its vertices point outwards.
• A concave polygon has at least one interior angle greater than — a reflex angle. It has a “dent”.

Two practical tests to teach:

1. The rubber-band test. Stretch a band around the shape. If it touches every vertex, the shape is convex. If some vertex sits inside the band, it is concave.
2. The diagonal test. In a convex polygon, every diagonal lies entirely inside. In a concave one, at least one diagonal passes outside.

Note: every regular polygon is convex. A concave polygon can never be regular, because a reflex interior angle cannot equal the others in a closed shape.

You do: Sketch each, or explain why it is impossible.

1. A convex pentagon
2. A concave pentagon
3. A concave triangle
4. A regular concave hexagon
5. A concave quadrilateral

(Answers: 1. Straightforward. 2. An arrowhead-like shape with one dent. 3. Impossible — a triangle’s three angles total , so none can exceed it. 4. Impossible — regular polygons are always convex. 5. Possible — a “dart”, which is a non-convex kite.)

Activity 3 — Inquiry: Diagonals and Symmetry (12 min)

Pairs, ruler and regular polygon templates.

For each regular polygon, count the number of diagonals and the number of lines of symmetry. Record in a table and look for patterns.

Sides	Diagonals	Lines of symmetry

Socratic scaffolding for the diagonal pattern:

Prompt	Purpose
Start by counting carefully for a pentagon.	diagonals.
How many lines can you draw from one vertex?	To every vertex except itself and its two neighbours — so .
For a pentagon, that is?	from each vertex.
Multiply by the number of vertices.	. But the count is — what went wrong?
Each diagonal has two ends.	So every diagonal was counted twice. Halve it.
Write the rule.	.
Test on a hexagon.	✓
Now symmetry — what do you notice?	A regular -gon has exactly lines of symmetry.

Completed table:

Sides	Diagonals	Lines of symmetry

Discussion: A triangle has no diagonals. Why? (Every pair of vertices is already joined by a side — there is nothing left to connect.)

Extension: How many diagonals does a -sided polygon have? (.)

Checks for Understanding

(5 minutes — exit ticket)

1. Name the polygon with (a) sides (b) sides (c) sides.
2. Is a rhombus a regular polygon? Explain.
3. Reasoning. Explain why a triangle cannot be concave.
4. How many diagonals does a regular octagon have?
5. How many lines of symmetry does a regular decagon have?

Answers: 1. (a) heptagon (b) nonagon (c) dodecagon; 2. No — its sides are all equal but its angles are not, and regularity needs both; 3. A concave polygon needs an interior angle greater than , but a triangle’s three angles total only ; 4. ; 5. .

Common Misconceptions

Misconception	How to pre-empt it
Believing equal sides alone make a polygon regular.	The rhombus counterexample, used every time regularity is discussed.
Believing equal angles alone make a polygon regular.	The rectangle counterexample, alongside it.
Thinking a polygon must be convex.	Sketch concave examples deliberately; use the rubber-band test.
Confusing sides with diagonals.	A side joins adjacent vertices; a diagonal joins non-adjacent ones.
Double-counting diagonals.	The inquiry derives the halving explicitly.
Assuming a shape drawn irregularly cannot be regular.	Regularity depends on measurements, not the neatness of the sketch.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). How many diagonals does a regular decagon have?

Answer

E2 (AMC Junior style). A polygon has diagonals. How many sides does it have?

Answer

Testing: gives ✓. It is a heptagon.

E3 (Challenge). A regular polygon has lines of symmetry. How many diagonals does it have?

Answer

Nine lines of symmetry means , a nonagon. Diagonals .

E4 (Challenge). How many triangles can be formed by joining three vertices of a regular hexagon?

Answer

Choosing vertices from :

Since no three vertices of a convex polygon are collinear, all give genuine triangles.

E5 (Reasoning challenge). Explain why a regular polygon can never be concave.

Answer

In a regular polygon all interior angles are equal. If one were reflex (greater than ), all would be — but then the total would far exceed the interior angle sum of any polygon, and the shape could not close. So every interior angle must be less than , making it convex.

Homework

1. Name the polygon with (a) sides (b) sides (c) sides (d) sides (e) sides.
2. State whether each is regular or irregular, giving a reason: (a) a square (b) a rectangle cm by cm (c) an equilateral triangle (d) a rhombus with a angle (e) a stop sign.
3. Calculate the number of diagonals in a regular: (a) pentagon (b) hexagon (c) nonagon (d) dodecagon.
4. State the number of lines of symmetry of a regular: (a) triangle (b) pentagon (c) octagon (d) dodecagon.
5. Sketch (a) a concave pentagon (b) a convex hexagon (c) a concave quadrilateral.
6. Reasoning. Explain why a rectangle is not a regular polygon even though all its angles are equal.
7. Reasoning. Explain why the number of lines of symmetry of a regular polygon always equals its number of sides.
8. A polygon has diagonals. How many sides has it?
9. Challenge. How many diagonals does a regular polygon with sides have?
10. Challenge. Explain why the formula gives for a triangle, and why that is correct.

Answers: Q1 — (a) hexagon (b) octagon (c) decagon (d) hendecagon (e) pentagon. Q2 — (a) regular (b) irregular, sides unequal (c) regular (d) irregular, angles unequal (e) regular octagon. Q3 — (a) (b) (c) (d) . Q4 — (a) (b) (c) (d) . Q6 — regularity requires equal sides as well, and a rectangle’s adjacent sides differ. Q7 — each line passes either through two opposite vertices or through two opposite edge midpoints, and there are exactly such lines. Q8 — , so , an octagon. Q9 — . Q10 — with , : from each vertex there are no non-adjacent vertices to join, since every other vertex is already a neighbour.