From an article in Mathematics Teaching 210 by Paul Stephenson

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Presentation transcript:

From an article in Mathematics Teaching 210 by Paul Stephenson How Many Sides? From an article in Mathematics Teaching 210 by Paul Stephenson Source: MT210 September 2008 p48 Paul Stephenson www.atm.org.uk/mt210

How Many Sides? How many sides has the polygon? (Not to scale) The diagram shows part of a regular polygon and part of some diagonals of the polygon drawn from one vertex. Two angles are marked. How many sides has the polygon? 20° 15°

How Many Sides? A polygon with 𝑛 sides (often called an n-gon)

How Many Sides? the external angle = 360° 𝑛 𝛼

How Many Sides? first diagonal 1 2 𝛼 𝛼

How Many Sides? 𝛼 2𝛼 𝛼

How Many Sides? second diagonal 𝛼 𝛼 2𝛼

How Many Sides? 𝛼 2𝛼 3𝛼

How Many Sides? 3 2 𝛼 third diagonal 3 2 𝛼 3𝛼

How Many Sides? 𝛼 3𝛼 4𝛼

How Many Sides? 2𝛼 fourth diagonal 2𝛼 4𝛼

How Many Sides? 𝛼 4𝛼 5𝛼

How Many Sides? 5 2 𝛼 fifth diagonal 5 2 𝛼 5𝛼

How Many Sides? 5 2 𝛼 These angles differ by 1 2 𝛼 each step, i.e. half the external angle. 2𝛼 3 2 𝛼 So determine the external angle, 𝛼, then determine the number of sides of the n-gon using: 𝑛= 360° 𝛼 𝛼 1 2 𝛼 ∙ Note that all the angles marked equal 𝛼 2 . 𝛼

Note to Teacher The difference between the angles shown is equal to half the external angle So 360 2×𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒 =number of sides E.g. if the angles are 16⁰ and 12⁰ Then the difference is 4⁰ So number of sides = 360° 2×4° =45

RESOURCES

How Many Sides? How many sides has the polygon? (Not to scale) The diagram shows part of a regular polygon and part of some diagonals of the polygon drawn from one vertex. Two angles are marked. How many sides has the polygon? 16° 12° Answer = 45 sides SIC_47

How Many Sides? How many sides has the polygon? (Not to scale) The diagram shows part of a regular polygon and part of some diagonals of the polygon drawn from one vertex. Two angles are marked. How many sides has the polygon? 20° 15° SIC_47

How Many Sides? How many sides has the polygon? (Not to scale) The diagram shows part of a regular polygon and part of some diagonals of the polygon drawn from one vertex. Two angles are marked. How many sides has the polygon? 24° 18° SIC_47

How Many Sides? How many sides has the polygon? (Not to scale) The diagram shows part of a regular polygon and part of some diagonals of the polygon drawn from one vertex. Two angles are marked. How many sides has the polygon? 36° 27° SIC_47

How Many Sides? How many sides has the polygon? (Not to scale) The diagram shows part of a regular polygon and part of some diagonals of the polygon drawn from one vertex. Two angles are marked. How many sides has the polygon? 30° 20° SIC_47

How Many Sides? How many sides has the polygon? (Not to scale) The diagram shows part of a regular polygon and part of some diagonals of the polygon drawn from one vertex. Two angles are marked. How many sides has the polygon? 48° 36° SIC_47

How Many Sides? How many sides has the polygon? (Not to scale) The diagram shows part of a regular polygon and part of some diagonals of the polygon drawn from one vertex. Two angles are marked. How many sides has the polygon? 45° 30° SIC_47

How Many Sides? How many sides has the polygon? (Not to scale) The diagram shows part of a regular polygon and part of some diagonals of the polygon drawn from one vertex. Two angles are marked. How many sides has the polygon? 54° 36° SIC_47

How Many Sides? How many sides has the polygon? (Not to scale) The diagram shows part of a regular polygon and part of some diagonals of the polygon drawn from one vertex. Two angles are marked. How many sides has the polygon? 28° 24° Answer = 45 sides SIC_47

How Many Sides? How many sides has the polygon? (Not to scale) The diagram shows part of a regular polygon and part of some diagonals of the polygon drawn from one vertex. Two angles are marked. How many sides has the polygon? 35° 30° SIC_47

How Many Sides? How many sides has the polygon? (Not to scale) The diagram shows part of a regular polygon and part of some diagonals of the polygon drawn from one vertex. Two angles are marked. How many sides has the polygon? 42° 36° SIC_47

How Many Sides? How many sides has the polygon? (Not to scale) The diagram shows part of a regular polygon and part of some diagonals of the polygon drawn from one vertex. Two angles are marked. How many sides has the polygon? 45° 36° SIC_47

How Many Sides? How many sides has the polygon? (Not to scale) The diagram shows part of a regular polygon and part of some diagonals of the polygon drawn from one vertex. Two angles are marked. How many sides has the polygon? 50° 40° SIC_47

How Many Sides? How many sides has the polygon? (Not to scale) The diagram shows part of a regular polygon and part of some diagonals of the polygon drawn from one vertex. Two angles are marked. How many sides has the polygon? 72° 60° SIC_47

How Many Sides? How many sides has the polygon? (Not to scale) The diagram shows part of a regular polygon and part of some diagonals of the polygon drawn from one vertex. Two angles are marked. How many sides has the polygon? 75° 60° SIC_47

How Many Sides? How many sides has the polygon? (Not to scale) The diagram shows part of a regular polygon and part of some diagonals of the polygon drawn from one vertex. Two angles are marked. How many sides has the polygon? 72° 54° SIC_47