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Polygons builds on earlier work with angle facts by extending these skills to interior and exterior angles of polygons. This unit develops the skill of using the angle sums and formulae for regular polygons.
Students learn to construct a regular polygon from a centre, then discover interior and exterior angles properties using polygons constructed within a circle. They then find the number of sides from a given interior angle, leaving circle theorems for later units.
By the end of this unit students will be able to:
Students should be secure with the following before beginning this unit.
A regular polygon has equal sides and equal angles. An irregular pentagon still has interior angles summing to 540°, but those angles are not all 108°.
A polygon with n sides splits into n − 2 triangles from one vertex. The interior sum is therefore (n − 2) × 180°. A pentagon is 3 × 180° = 540°, not 5 × 180°.
The exterior angles of any polygon, taken the same way around, sum to 360°. For a regular n-gon each exterior angle is 360° ÷ n.
At each vertex the interior angle and the adjacent exterior angle sit on a straight line, so they sum to 180°. Finding one finds the other.
| Misconception | Teaching focus |
|---|---|
| Calculating the interior sum as n × 180°, so a pentagon becomes 900°. | Split the pentagon into triangles from one vertex and count the triangles before the formula is written. |
| Assuming every pentagon has interior angles of 108°, including an irregular one. | Keep an irregular pentagon on the board and only use 540° as a total, not as five equal angles. |
| Thinking the exterior-angle sum grows with the number of sides, so a hexagon has a larger exterior total than a triangle. | Walk around any closed shape and insist the turning totals 360°, whatever n is. |
| Adding exterior angles that do not all turn the same way, so the total is not 360°. | Mark every exterior angle on the outside of the shape, all clockwise, before any adding starts. |
| Using 360° ÷ n for an interior angle, or forgetting that interior and exterior at a vertex sum to 180°. | Write exterior first from 360° ÷ n, then subtract from 180° for the interior of a regular polygon. |
Every lesson in this unit is already built. A Mr Mathematics membership gives you the presentation, worksheet, question generator and video tutorial for both lessons, alongside the full Key Stage 3, GCSE, IGCSE and A-Level libraries.
A school membership covers every teacher in your department, so the whole scheme of work is resourced from one subscription.
Students learn how to construct regular polygons using a pair of compasses.
Later, as learning progresses, they investigate the relationship between interior and exterior angles.
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Students use the properties of regular polygons to calculate interior and exterior angles.
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Students learn how to solve a range of problems involving polygons, focusing on interior and exterior angles, angle sums, symmetry and polygon properties.
As the lesson progresses, students apply these ideas to more challenging problem-solving and reasoning questions. They move from using known angle facts to determining the number of sides in polygons.
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Students revise calculating the interior and exterior angles of regular polygons.
As learning progresses, they apply multiple-angle properties to solve more complex geometrical problems.
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