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# Polygons

Angles and shapes · Classifying shapes

<!-- Speaker note: Students change the number of sides and drag corners to see polygons named (pentagon to decagon), regular or irregular, and convex or concave, and reason about why equal sides alone are not enough (AC9M7SP02, node 3). About 35 minutes (Predict and Run 8 min · Investigate 10 min · Modify 8 min · Practise and Make 9 min). -->

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## Today

**We are learning to** a polygon is named by its number of sides (pentagon 5, hexagon 6, octagon 8, decagon 10). It is regular when all its sides and all its angles are equal, and concave when a corner points in.

- I can name polygons from 3 to 10 sides.
- I can tell whether a polygon is regular, and explain why.
- I can tell a convex polygon from a concave one.

<!-- Speaker note: Before class: Do the Run board yourself first. Students need the quadrilateral family (previous lesson) and the polygon angle sum (Angles in a polygon). The one idea: regular means equal sides **and** equal angles; concave means a corner points in. The common mistake is equal sides alone (Kim's mistake with a rhombus). A dart (a kite with one corner pushed in) is named a kite by the board, as its sides fit the kite's definition; some textbooks keep kites convex and call the concave one a dart. The board starts each polygon regular (its corners are off the grid); dragging a corner puts it on the grid, so it becomes irregular. -->

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## Why it matters

> Stop signs are octagons, honeycomb cells are hexagons, and a star is a polygon with corners that point in. What makes some of them regular?

<!-- Speaker note: A **polygon** is a closed shape with straight sides. Its name counts the sides: | Sides | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |---|---|---|---|---|---|---|---|---| | Name | triangle | quadrilateral | pentagon | hexagon | heptagon | octagon | nonagon | decagon | A polygon is **regular** when **all its sides are equal and all its angles are equal**; otherwise it is **irregular**. It is **convex** when every corner points out (every angle less than 180°), and **concave** when at least one corner points in (an angle more than 180°). -->

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<span style="background:#dbeafe; color:#1a1a1a; border:1.5px solid #3b82f6; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">💬 Discussion</span>

## Predict

**A hexagon has six equal sides. Must it be a regular hexagon?**

- Only if it is convex
- No: its angles might not be equal
- Yes: equal sides make it regular

<!-- Speaker note: What happens: No. Squash a regular hexagon and the sides stay equal but the angles change. Regular needs equal sides and equal angles. Take two or three predictions before anyone runs it. -->

<!-- Speaker note: Teacher: ask a 'yes' student to imagine squashing a regular hexagon. -->

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## Try it

{{embed: https://studio.learningwaypoints.com/tutorials/_playground-board/index.html?id=deck%2Fmaths%2Fangles%2Fpolygons%2Fstep-2&code=widget%3A+geometry%0Ashape%3A+polygon%0Asides%3A+choose%0Aclassify%3A+on%0Asum%3A+off%0A&fresh=1}}

[Open this step in the tutorial](https://studio.learningwaypoints.com/tutorials/maths/angles/polygons/?full=step-2#step-2)

<!-- Speaker note: **More sides** and **Fewer sides** change the shape; it starts regular. Then drag a corner. -->

<!-- Speaker note: Teacher: ask what happens to one angle when its corner points in. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## What happened?

* No. Squash a regular hexagon and the sides stay equal but the angles change. Regular needs equal sides and equal angles.

<!-- Speaker note: Reveal after running. Ask who predicted it, and what made them think so. -->

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<span style="background:#dbeafe; color:#1a1a1a; border:1.5px solid #3b82f6; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">💬 Discussion</span>

## Count, then check

Here is the routine for naming a polygon.

Now match the names.

$$\text{6 sides} \Rightarrow \text{hexagon}$$

1) $\text{sides equal and angles equal?}$ — Both: regular. Either one fails: irregular.
2) $\text{an angle} > 180^\circ?$ — Yes: concave (a corner points in). No: convex.
3) $\text{square} = \text{regular quadrilateral}$ — A rhombus has equal sides but not equal angles; a rectangle the reverse.

<!-- Speaker note: Model this live on the board before revealing anything: write each line and say why, pausing to ask the class what comes next. Then click to reveal the lines and compare. Script. Start: write 6 sides hexagon and say "Count the sides for the name." Line 1: write sides equal and angles equal? and say "Both: regular. Either one fails: irregular." Line 2: write an angle > 180^? and say "Yes: concave (a corner points in). No: convex." Line 3: write square = regular quadrilateral and say "A rhombus has equal sides but not equal angles; a rectangle the reverse." -->

<!-- Speaker note: Teacher: link the prefixes to words they know (octopus, decade, pentathlon). -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Count, then check

**Match each polygon to its number of sides.**

| | |
|---|---|
| `pentagon` | A. 7 |
| `hexagon` | B. 6 |
| `heptagon` | C. 10 |
| `octagon` | D. 8 |
| `decagon` | E. 5 |

* Answer: pentagon → E. 5; hexagon → B. 6; heptagon → A. 7; octagon → D. 8; decagon → C. 10

<!-- Speaker note: Pent- five, hex- six, hept- seven, oct- eight (an octopus has eight arms), dec- ten. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Find the mistake

**Kim decides whether a rhombus is regular. Click the line where her reasoning goes wrong.**

```
 1  This rhombus has four sides of 5 cm.
 2  Its angles are 60°, 120°, 60° and 120°.
 3  All its sides are equal, so it is a regular quadrilateral.
```

* Answer: line 3

<!-- Speaker note: Line 3: regular needs equal angles too, and 60° and 120° are not equal. The only regular quadrilateral is a square. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Concave

**How can you tell that a polygon is concave?**

- At least one corner points in, with an angle more than 180°
- It has a right angle
- It has more than six sides
- Its sides are not all equal

* Answer: At least one corner points in, with an angle more than 180°

<!-- Speaker note: A concave polygon has a corner that points in: a reflex angle, over 180°. -->

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<span style="background:#fef3c7; color:#1a1a1a; border:1.5px solid #f59e0b; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✍️ Activity</span>

## One change at a time



- **A concave pentagon:** Each task changes one thing.
- **An irregular octagon:** 
- **Rectangles:** 
- **Circles:** 

[Open this step](https://studio.learningwaypoints.com/tutorials/maths/angles/polygons/?full=step-4#step-4)

<!-- Speaker note: What this step is for: One change at a time on the board: a concave pentagon, an irregular convex octagon, then why a rectangle is not regular and whether a circle is a polygon. Hints, in order: One More sides click, then drag one corner towards the middle. / Make 8 sides first: it starts regular. / Then move one corner a little, keeping every corner pointing out. For students who need more support: Keep the names table visible; tick 'sides equal' and 'angles equal' separately for each shape. For early finishers: Which regular polygons tile a floor with no gaps? Use their angles (60°, 90°, 108°, 120°) to explain. The hard part, on purpose: Checking both sides and angles before calling a shape regular. Ask students to say what they checked for each, rather than confirming their answer. When students get stuck: A student calls a rhombus regular (Kim's mistake). Ask: Are its angles all the same? Fix: Regular needs equal sides and equal angles. / A student mixes up convex and concave. Ask: Does any corner point into the shape, like a cave? Fix: Concave has a 'cave': a corner pointing in. / A student confuses heptagon and hexagon. Ask: Which has six: hex or hept? Fix: Hex six, hept seven (like September, the old seventh month). -->

<!-- Speaker note: Teacher: for the octagon, ask how small a move breaks regular. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Names

**What is a polygon with 9 sides called?**

- Heptagon
- Octagon
- Decagon
- Nonagon

* Answer: Nonagon

<!-- Speaker note: 9 sides: a nonagon. -->

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## Regular

**Which is a regular polygon?**

- A rhombus with angles 70° and 110°
- An equilateral triangle
- A rectangle 4 cm by 6 cm
- An isosceles triangle

* Answer: An equilateral triangle

<!-- Speaker note: An equilateral triangle has equal sides and equal angles (60° each). -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## A regular hexagon

**Each angle of a regular hexagon is the same. The angles add to 720°. How big is each one, in degrees?**

* Answer: 120

<!-- Speaker note: 720 ÷ 6 = 120°. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Review: angle sums

**Review: what do the angles of an octagon add to, in degrees?**

* Answer: 1080

<!-- Speaker note: (8 − 2) × 180 = 1080°. -->

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<span style="background:#fef3c7; color:#1a1a1a; border:1.5px solid #f59e0b; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✍️ Activity</span>

## Make: a dart

Then: is your dart regular? Why can a concave polygon never be regular?

[Open this step](https://studio.learningwaypoints.com/tutorials/maths/angles/polygons/?full=step-6#step-6)

<!-- Speaker note: What this step is for: Building a concave kite (a dart) combines a family name with concave: the kite's properties survive a corner pointing in. Hints, in order: A kite has two pairs of equal sides next to each other. / Push one corner of a kite in past the middle, like an arrowhead. -->

<!-- Speaker note: Teacher: ask whether a dart is still a kite, and why. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Exit ticket

**A pentagon has angles 100°, 110°, 110°, 100° and 120°. Is it regular? Convex or concave?**

* Irregular (its angles are not equal), convex (every angle under 180°).

<!-- Speaker note: Students answer on paper or a mini whiteboard before the reveal. Anyone who cannot answer needs the main idea again next lesson. Where this leads: This completes AC9M7SP02. Where it leads: tessellations, and the interior and exterior angles of regular polygons (Year 8); Pythagoras (AC9M8M06) for right-angled triangles. -->