The shape of the lesson
By the end of the lesson, learners can
- The angle sum: In the polygons tab, change the number of sides and watch the polygon split into triangles from one vertex.
- Exterior angles and regular polygons: Imagine walking round the pentagon in Fig. 1.
- Polygons together: Questions often fit polygons together at a point or along an edge.
- Angle sum: \((n - 2) \times 180^\circ\).
- Exterior sum: always \(360^\circ\).
- Regular: exterior \(= 360^\circ \div n\), interior \(= 180^\circ - \text{exterior}\).
- Number of sides: \(n = 360^\circ \div \text{exterior}\); it must be a whole number.
- Shapes at a point: interior angles add to \(360^\circ\).
One picture every idea returns to
The exterior angles of any polygon add up to \(360^\circ\), and at each vertex interior + exterior \(= 180^\circ\). The interior angles of an \(n\)-sided polygon add up to \((n - 2) \times 180^\circ\). For a regular polygon, each exterior angle is \(\dfrac{360^\circ}{n}\).
Forty-five minutes, phase by phase
| Time | Phase | What happens in the room | Grouping |
|---|---|---|---|
| 0 to 5 min | Hook: predict | Open with the question in the title, Walk round, turn 360°.. Walk all the way round any polygon and you turn through exactly one full turn. Put the figure on the board: it is the one picture every idea in the lesson returns to. | Whole class, then pairs |
| 5 to 13 min | Teach and check: The angle sum | In the polygons tab, change the number of sides and watch the polygon split into triangles from one vertex. Project tab 5 of the lab (Angles lab, polygons) and let a student drive it while the class predicts. Model the worked example A missing angle (Extended: irregular polygon) on the board, one line at a time. | Whole class, then pairs |
| 13 to 21 min | Teach and check: Exterior angles and regular polygons | Imagine walking round the pentagon in Fig. 1. Model the worked example Interior angle of a regular polygon on the board, one line at a time. Your turn: Can a regular polygon have interior angles of \(130^\circ\)? Explain. | Pairs |
| 21 to 30 min | Teach and check: Polygons together | Questions often fit polygons together at a point or along an edge. Model the worked example A square and a hexagon on the board, one line at a time. Your turn: Three regular pentagons are placed round a point, edge to edge. What angle is left uncovered? | Rally Coach |
| 30 to 36 min | Check | The four Stage 1 questions from the student lesson, answered on mini whiteboards or by show of fingers. Re-teach the one most get wrong. | Individual, then whole class |
| 36 to 43 min | Plenary | The exam-style questions from Stage 2, one at a time: thirty seconds alone, then a random call. The feedback lines in the key give the examiner's wording. | Individual, random call |
| 43 to 45 min | Exit and homework | Set the worksheet below (6 questions, 18 marks). Learners self-assess against the objectives before they leave. | Individual |
Protect the parts that carry the learning
The sequence carries about fifty minutes of material, so the lab is driven by one student while the class predicts rather than explored freely. Times are a guide.
Protected: the check at 30 minutes. It is the only individual read on whether the key idea landed, and it is not cut.
Cut first if the build overruns: the last check in Polygons together, never the check or the plenary.
In a 60 minute block: run worksheet questions 1 to 3 in class as a Rally Coach set and model both the hardest worked example and its check live.
Rally Coach
Pairs share one worksheet. One partner solves the first question aloud while the other coaches, checking each line and offering a prompt rather than the answer; they swap roles for the next question.
The full structure, with its variations and the other twelve, is on the cooperative-learning page.
What to head off, and how
| Trap learners fall into | Teaching move that pre-empts it |
|---|---|
| The formula is \((n - 2) \times 180^\circ\), not \(n \times 180^\circ\). | Splitting from one vertex gives \(n - 2\) triangles; a pentagon makes 3 triangles, not 5. |
| Dividing 360 by the INTERIOR angle is a common slip: \(360 \div 156\) is not a number of sides. | Convert to the exterior angle first. |
| Do not use the exterior angle when shapes meet at a point. | The corners that fill the space round a point are the INTERIOR angles. |
Angles in polygons: worksheet
6 questions, 18 marks. Written in the style of the papers; no past paper question is reproduced.
(a) Find the sum of the interior angles of an octagon. (b) Find each interior angle of a regular octagon.
Each exterior angle of a regular polygon is \(20^\circ\). Find (a) the number of sides (b) each interior angle (c) the sum of the interior angles.
Show that a regular polygon can have interior angles of \(140^\circ\) but not \(145^\circ\).
(Extended) The interior angles of a hexagon are \(130^\circ\), \(125^\circ\), \(140^\circ\), \(85^\circ\), \(x\) and \(2x\). Find the largest angle.
Explain why the interior angles of a polygon with \(n\) sides add up to \((n - 2) \times 180^\circ\).
A floor is tiled with regular octagons and squares, with two octagons and one square meeting at every corner.
(a) Find the interior angle of a regular octagon. (b) Show that two octagons and one square fit exactly round a point. (c) Explain why regular pentagons alone cannot tile a floor.
Answer key and marking notes
- (a) \((8 - 2) \times 180^\circ = 1080^\circ\).
- (b) \(1080^\circ \div 8 = 135^\circ\) (or \(180^\circ - 360^\circ \div 8 = 135^\circ\)).
- (a) \(360 \div 20 = 18\) sides.
- (b) \(180^\circ - 20^\circ = 160^\circ\).
- (c) \((18 - 2) \times 180^\circ = 2880^\circ\) (check: \(18 \times 160 = 2880\)).
- \(140^\circ\): exterior \(= 40^\circ\), \(n = 360 \div 40 = 9\), a whole number, so a regular nonagon works.
- \(145^\circ\): exterior \(= 35^\circ\), \(n = 360 \div 35 = 10.28...\), not a whole number, so it is impossible.
- Hexagon angle sum \(= 4 \times 180^\circ = 720^\circ\).
- \(130 + 125 + 140 + 85 + 3x = 720\), so \(480 + 3x = 720\) and \(x = 80\).
- The angles \(x\) and \(2x\) are \(80^\circ\) and \(160^\circ\); the largest is \(160^\circ\).
- Join one vertex to every other vertex that is not next to it.
- This splits the polygon into \(n - 2\) triangles (each side except the two at the chosen vertex forms one triangle).
- The triangles' angles together make up exactly the polygon's interior angles, and each triangle contributes \(180^\circ\), so the sum is \((n - 2) \times 180^\circ\).
- (a) Exterior \(= 360 \div 8 = 45^\circ\), so interior \(= 135^\circ\).
- (b) \(135^\circ + 135^\circ + 90^\circ = 360^\circ\), exactly the angles at a point, so there is no gap or overlap.
- (c) Each interior angle of a regular pentagon is \(108^\circ\). \(360 \div 108 = 3.33...\) is not a whole number, so pentagons leave a gap (\(36^\circ\) with three) or overlap (with four).
6 multiple-choice questions on this lesson
From the topic question bank. Assign them in the app for a marked score in your gradebook, or run the topic as a class game.
Find the sum of the interior angles of a nonagon (9 sides).
- A\(1260^\circ\)
- B\(1620^\circ\)
- C\(1440^\circ\)
- D\(140^\circ\)
Each interior angle of a regular polygon is \(162^\circ\). How many sides does it have?
- A20
- B2.22
- C18
- D10
Find the size of each interior angle of a regular 15-sided polygon.
- A\(156^\circ\)
- B\(24^\circ\)
- C\(168^\circ\)
- D\(2340^\circ\)
The diagram shows a regular pentagon and a square that share a side. Find the angle marked \(x\).
- A\(162^\circ\)
- B\(198^\circ\)
- C\(18^\circ\)
- D\(252^\circ\)
The interior angles of a pentagon are \(3x\), \(2x + 10^\circ\), \(4x - 20^\circ\), \(100^\circ\) and \(135^\circ\). Find \(x\).
- A\(35\)
- B\(15\)
- C\(75\)
- D\(55\)
Each interior angle of a regular polygon is 4 times the size of each exterior angle. How many sides does the polygon have?
- A10
- B5
- C8
- D36
Quiz key
From one vertex a 9-sided polygon splits into \(9 - 2 = 7\) triangles, so the angle sum is \(7 \times 180^\circ = 1260^\circ\). The tempting \(1620^\circ\) uses \(9 \times 180^\circ\), forgetting that only \(n - 2\) triangles fit. \(1440^\circ\) is the decagon's sum, and \(140^\circ\) is ONE interior angle of a REGULAR nonagon.
Exterior angle \(= 180^\circ - 162^\circ = 18^\circ\), so \(n = 360^\circ \div 18^\circ = 20\). The tempting 2.22 divides \(360^\circ\) by the INTERIOR angle, which is not a whole number of sides. 18 stops at the exterior angle, and 10 divides \(180^\circ\) by \(18^\circ\).
Exterior angle \(= 360^\circ \div 15 = 24^\circ\), so each interior angle is \(180^\circ - 24^\circ = 156^\circ\). Check: \(13 \times 180^\circ \div 15 = 156^\circ\). The tempting \(24^\circ\) is the exterior angle. \(168^\circ\) divides \((15 - 1) \times 180^\circ\) by 15, and \(2340^\circ\) is the sum of all the interior angles.
The pentagon's interior angle is \(540^\circ \div 5 = 108^\circ\) and the square's is \(90^\circ\). Angles at a point: \(x = 360^\circ - 108^\circ - 90^\circ = 162^\circ\). The tempting \(198^\circ\) uses the pentagon's EXTERIOR angle \(72^\circ\) instead of its interior angle. \(18^\circ\) is \(108^\circ - 90^\circ\), and \(252^\circ\) leaves out the square.
Pentagon angle sum \(= 3 \times 180^\circ = 540^\circ\). \(3x + 2x + 10 + 4x - 20 + 100 + 135 = 540\), so \(9x + 225 = 540\), \(9x = 315\) and \(x = 35\). The tempting 15 uses \(360^\circ\), the quadrilateral's sum. 75 uses \(5 \times 180^\circ = 900^\circ\), and 55 uses the hexagon's \(720^\circ\).
Interior + exterior \(= 180^\circ\), so \(4e + e = 180^\circ\), \(e = 36^\circ\), and \(n = 360^\circ \div 36^\circ = 10\). The tempting 5 sets \(4e + e = 360^\circ\), confusing one pair of angles with the sum of all the exterior angles. 8 is \(n - 2\) (from \(1440^\circ \div 180^\circ\)), and 36 is the exterior angle, not the number of sides.
What to project
The student lesson embeds these tabs. Open them full screen on the projector; every slider has a drag handle on the stage.
Slides for this lesson
A projectable deck for every pack is on the way. Until then, present the student lesson full screen: its sections, figure and lab tabs follow the same order as the plan above.