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.
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?
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°).
Predict: Read the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.
Try: Try it on the board and compare the result with your prediction.
Investigate: Work out why it happens, one step at a time.
Modify: Change the problem on purpose and see what happens.
Practise: A few on your own, to check you have it.
Make: Make your own problem using what you have learned. Expect a few tries; that is normal.
Make your prediction first. Help opens once you've locked it in.
PredictRead the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.
Six equal sides
Picture pushing two opposite corners of a regular hexagon towards each other.
TryTry it on the board and compare the result with your prediction.
A corner that points in
More sides and Fewer sides change the shape; it starts regular. Then drag a corner.
Try it yourselfYour edits are saved in this browser.
Make a hexagon with one corner pointing in (concave). Is it regular? Was your prediction about sides and angles right?
Use the board. This ticks when you get there.
InvestigateWork out why it happens, one step at a time.
Names and properties
Here is the routine for naming a polygon.
Worked exampleBefore each step, say what you think comes next.
6 sides⇒hexagonCount the sides for the name.
sides equal and angles equal?Both: regular. Either one fails: irregular.
an angle>180∘?Yes: concave (a corner points in). No: convex.
square=regular quadrilateralA rhombus has equal sides but not equal angles; a rectangle the reverse.
Now match the names.
Read Kim's reasoning. What else must be equal?
Match each polygon to its number of sides.
pentagon
hexagon
heptagon
octagon
decagon
Kim decides whether a rhombus is regular. Click the line where her reasoning goes wrong.
ModifyChange the problem on purpose and see what happens.
One change at a time
Each task changes one thing.
Need a hint?
One More sides click, then drag one corner towards the middle.
Need a hint?
Make 8 sides first: it starts regular.
Another hint
Then move one corner a little, keeping every corner pointing out.
Try it yourselfYour edits are saved in this browser.
Make a concave pentagon.
Use the board. This ticks when you get there.
Make an octagon that is convex but not regular.
Use the board. This ticks when you get there.
PractiseA few on your own, to check you have it.
Polygons
Each angle of a regular hexagon is the same. The angles add to 720°. How big is each one, in degrees?
Review: what do the angles of an octagon add to, in degrees?
MakeMake your own problem using what you have learned. Expect a few tries; that is normal.
A dart
Then: is your dart regular? Why can a concave polygon never be regular?
Try it yourselfYour edits are saved in this browser.
Need a hint?
A kite has two pairs of equal sides next to each other.
Another hint
Push one corner of a kite in past the middle, like an arrowhead.
Make a dart: a kite with one corner pointing in.
Use the board. This ticks when you get there.
For teachers
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.
I can tell whether a polygon is regular, and explain why.
I can tell a convex polygon from a concave one.
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.
Running the lesson
Predict: ask a 'yes' student to imagine squashing a regular hexagon.
Run: ask what happens to one angle when its corner points in.
Investigate: link the prefixes to words they know (octopus, decade, pentathlon).
Modify: for the octagon, ask how small a move breaks regular.
Make: ask whether a dart is still a kite, and why.
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).
Stretch and support
Stretch: Which regular polygons tile a floor with no gaps? Use their angles (60°, 90°, 108°, 120°) to explain.
Support: Keep the names table visible; tick 'sides equal' and 'angles equal' separately for each shape.
Where it 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.
What each step is for
six equal sides: Committing to whether equal sides alone make a regular hexagon shows whether they know regular needs equal angles too.
a corner that points in: Changing the number of sides and dragging one corner inwards shows regular, irregular and concave as properties of any polygon.
names and properties: Names count sides; regular needs equal sides and equal angles (Kim checks only the sides); concave means a corner points in.
one change at a time: 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.
polygons: Names, regular or not and an angle sum, mixed; equal sides alone is the planted slip; the review keeps the polygon angle sum fresh.
a dart: Building a concave kite (a dart) combines a family name with concave: the kite's properties survive a corner pointing in.
Now you can
Name polygons from 3 to 10 sides.
Tell whether a polygon is regular, and explain why.