At a glance. One lesson of about 60 minutes. Students find the perimeter and area of a regular polygon from its side and apothem, explain why the area is n triangles, and work backwards to a missing apothem.
Calculator: scientific calculator. Apothems are rounded decimals, so the products are not easy to do in the head. The calculator is for the arithmetic and the structure stays with the student.
What students will learn
Know
- A regular polygon has equal sides and equal angles.
- The apothem is the distance from the centre to the middle of a side, at a right angle to it.
- Area = n × ½ × s × a, which is ½ × perimeter × a.
Understand
- The polygon is n identical triangles with base s and height a.
- The triangles can be laid in a row to make a parallelogram or a trapezium of the same area.
- The apothem is shorter than the distance from the centre to a corner.
Do
- Find the perimeter and area of a regular polygon.
- Use the angle at the centre, 360 ÷ n.
- Find a missing apothem, and use the perimeter when the number of sides is not given.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: apothem, the area formula and the angle at the centre (questions 1 to 9).
- Criterion B. Investigating patterns: change the number of sides and watch the apothem and area; explain what happens to a polygon with a fixed perimeter as the sides increase (the what-if question).
- Criterion C. Communicating: label the apothem with a square corner and write one triangle's area before multiplying (questions 4 and 5).
- Criterion D. Applying mathematics in real-life contexts: tiles and a floor, where the cost needs the area first (question 7 and the tile worked example).
The one thing to do with the figure
Make students count before the triangles appear. Set a heptagon and ask how many triangles will appear and what the angle at the centre of each will be. Then press Show the triangles.
In the second figure, set an odd number of sides and ask what shape the row will be. Most say a parallelogram. The slanted end that is not parallel is the new thing: it is a trapezium, and lesson 1 gave you its area.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 10 min | The tile and the prediction. First figure: show the triangles and read off the area. |
| 10 min | Key idea. Students draw a regular hexagon with a compass (radius equal to the side) and measure the apothem. |
| 10 min | Second figure, worked examples, then the faded one in pairs. |
| 5 min | Question 1, the check. If many choose C, go back to the first figure. |
| 15 min | Your turn, Levels 1 to 3. |
| 5 min | Exit ticket, question 9, on paper or on screen. |
The answers
| Warm-up | 72°; 20 cm²; 24 cm². |
|---|---|
| Worked examples | 261 cm²; 221.4 cm²; 120 cm². |
| Faded example | 172.5 cm². |
| 1. Which calculation gives the area | A. B forgets the half, C finds one triangle, D leaves out the apothem. If many choose C, go back to the first figure and press Show the triangles for a pentagon. |
| 2. Angle at the centre | 40°. 360 ÷ 9 = 40. |
| 3. Area of one triangle | 16.8 cm². ½ × 7 × 4.8 = 16.8. |
| 4. Area of a regular hexagon | 93.6 cm². 6 × 15.6 = 93.6. |
| 5. Area from a diagram | 276 cm². 10 × 27.6 = 276. |
| 6. Missing apothem | 12.1 cm. 484 ÷ 40 = 12.1. |
| 7. Area and cost | 5100 rupees. 10.2 × 500 = 5100. |
| 8. Perimeter instead of the sides | 374.4 cm². 36 × 10.4 = 374.4. |
| 9. Height of the triangle | A. B uses the corner distance as the height (the warning card's mistake), C forgets the half, D stops at one triangle. The side is longer than either figure allows. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| C in question 1 | Finds one triangle. | Ask how many triangles there are. Count them on the figure. |
| 3240 or 20 in question 2 | Multiplies, or uses a half turn. | Draw the angles meeting at the centre and ask what they add to. |
| 33.6 or 11.8 in question 3 | Forgets the half, or adds. | Draw the parallelogram round the triangle. |
| 187.2 or 15.6 in question 4 | Forgets the half, or stops at one triangle. | Ask for the area of one triangle in square centimetres first. |
| 552 or 27.6 in question 5 | Forgets the half, or stops at one triangle. | Ask how many triangles a decagon has. |
| 6.05 or 121 in question 6 | Halves twice, or loses the side. | Write 8 × ½ × 10 = 40 and ask what times 40 is 484. |
| 10200 or 850 in question 7 | Forgets the half, or prices one triangle. | Ask what the price is for. Square metres of floor. |
| 748.8 or 82.4 in question 8 | Forgets the half, or adds. | Say it aloud: half of the perimeter times the apothem. |
| B, C or D in question 9 | Uses the corner distance as the height, forgets the half, or stops at one triangle. | Draw one triangle and mark the square corner at the middle of the side. |
Questions worth asking
- Why are the triangles identical? All the sides are equal, and the angles at the centre are equal. Listen for “same base, same height, same angle”.
- Why is the apothem shorter than the line from the centre to a corner? The corner line is the longest side of a right-angled triangle that has the apothem as one side. Listen for “the corner is further away than the middle of a side”.
- What shape is the row when n is odd? A trapezium, with parallel sides that average half the perimeter.
- Do you need n to find the area? Not if you know the perimeter. Listen for “n times s is the perimeter”.
Differentiation
A way in for anyone stuck
- Give a printed regular polygon cut into triangles, so students can count and colour them.
- Give a set square, and have them mark the right angle at the foot of the apothem.
- Ask for the area of one triangle first and write it on its own line before multiplying.
Stretch
- A regular hexagon with side s has apothem about 0.87 × s. Show that its area is about 2.6 × s². (6 × ½ × s × 0.87 s = 2.61 s².)
- Compare 40 m of fence round a square, a regular hexagon and a regular octagon. Which encloses most? (Square 100 m², hexagon about 115 m², octagon about 121 m².)
Students learning in English as an additional language
“Regular” has two meanings in everyday English; here it means all sides and all angles are equal. “Apothem” is a new word: say “the line from the centre to the middle of a side, at a right angle” until it is familiar. “Polygon” means many-sided.
| Word | Plain meaning | Example |
|---|---|---|
| regular polygon | a flat shape with equal sides and equal angles | A square, or the face of a stop sign |
| apothem | the distance from the centre to the middle of a side, at a right angle to it | The height of one of the triangles |
| perimeter | the distance all the way round a shape | The fence round a field |
| centre | the point in the middle of a regular polygon | Where the triangles meet |
| hexagon | a polygon with six sides | A honeycomb cell |
| identical | exactly the same shape and size | The six triangles in a hexagon |
Sentence starters for explaining aloud or in writing:
- The polygon has … sides, so it is … identical triangles.
- The base of each triangle is … and the height is the apothem, …
- I did not need to know … because …
Two things not to say
“The apothem is the radius.” The radius of the circle round the polygon goes to a corner. The apothem goes to the middle of a side, at a right angle.
“Times by the number of sides, and that is the answer.” That is the perimeter or the area of n rectangles, not of n triangles.
Questions to set
Extra practice in three tiers, including one question from earlier work for spacing. Every answer has been checked.
1Fluency
The idea on its own, with friendly numbers. Set these first.
- FluencyA regular octagon has sides of 8 cm. Find its perimeter.
64 cm. - FluencyA triangle has a base of 8 cm and a height of 6.9 cm. Find its area.
27.6 cm².
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA tile in Marrakech is a regular octagon with sides of 5 cm and an apothem of 6 cm. How many square centimetres of glaze do 50 tiles need?
6000 cm²: 8 × ½ × 5 × 6 = 120 for one tile, and 50 × 120 = 6000. - SpacingFrom Year 8 Unit 3, lesson 5: what is (−4)²?
16.
3Challenge
Reasoning, generalizing and working backwards.
- MissingA regular pentagon has an area of 172.5 cm² and an apothem of 6.9 cm. How long is each side?
10 cm: 5 × ½ × s × 6.9 = 172.5 gives s = 172.5 ÷ 17.25 = 10. - ReasoningTwo regular polygons have the same apothem, 8 cm. One has a perimeter of 40 cm and the other 60 cm. What is the ratio of their areas?
2 : 3, the same as the ratio of the perimeters: 160 cm² and 240 cm².
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
Every figure works with the keyboard (tab to a slider, then the arrow keys) and by touch. Under each one, “Show me the steps” gives the same states in words.
No device? Squared paper, scissors, string and a few sheets of card do the same job: cut, turn, wrap and fold.