T Madas T Madas A regular heptagon 7

A regular heptagon (7 sided polygon) ABCDEFG is drawn below. Calculate the size of

A regular hexagon is shown below. Calculate the angles marked as x, y and

Three regular pentagons are placed in the way shown below so that they all

A regular polygon has exterior angle of 40°. How many sides does it have?

Part of a regular polygon is shown below. Its exterior angle is 20° How

The figure below shows a pentagon ABCDE. REAB = RABC = 105° and RBCD

A regular enneagon, (9 -sided polygon), is drawn below. Calculate the angles marked as

Calculate the exterior angle of a regular pentagon 360° ÷ 5 = 72° 180°

The diagram below shows part of a tessellation consisting of a regular octagon and

5° 13 135° The diagram below shows part of a tessellation consisting of a

A tessellation of squares and regular octagons is shown opposite. Calculate the angle marked

Regular dodecagons tessellate with equilateral triangles as shown opposite. Use this fact to calculate

The sides of a regular pentagon are extended until they form the star shape

Calculate the area of a regular hexagon of side 8 cm, giving your answer

There is a better method for this problem © T Madas

A regular enneagon (9 -sided polygon) ABCDEFGHI is shown below. Calculate RICE. I A

18 identical slabs in the shape of isosceles trapeziums fit tightly around a small

A regular octagon is inscribed in a circle of radius 5 cm. Calculate the

A regular decagon is inscribed in a circle of radius 4 cm. Calculate the

A regular hexagon and a regular pentagon are made to overlap each other as

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A regular heptagon (7 sided polygon) ABCDEFG is drawn below. Calculate the size of RAOC, giving your answer to 3 significant figures. A 360° ÷ 7 ≈ 51. 4286° B G 51. 4286° x 2 ≈ 103° [3 s. f. ] O F C E D © T Madas

A regular hexagon is shown below. Calculate the angles marked as x, y and z. 360° ÷ 6 = 60° 12 0° 30 z° 60° y 120° x 60° © T Madas

Three regular pentagons are placed in the way shown below so that they all share one vertex. Calculate the angle marked as x 54 ° 72 ° 54108° ° 108° x 36° © T Madas

A regular polygon has exterior angle of 40°. How many sides does it have? exterior angle = 360° n This is an easy problem… … but you are under exam pressure and you forgot this formula…. … in fact good mathematicians do not memorise this formula… … go back to basics © T Madas

A regular polygon has exterior angle of 40°. How many sides does it have? 40° 70° 360 ÷ 40 = 9 sides 40° © T Madas

Part of a regular polygon is shown below. Its exterior angle is 20° How many sides does it have? Calculate the angle marked as y. The exterior angle of an n-sided regular polygon is given by: 360° 20 ° n 20 ° 360 ÷ 20 = 18 sides y ° 20 20° The exterior angle of an n-sided regular polygon is equal to its central angle. y = 80° © T Madas

The figure below shows a pentagon ABCDE. REAB = RABC = 105° and RBCD = RDEA = 100°. Calculate the angle REDC. D x E … split the pentagon into 3 triangles … C 100° … the angles of a triangle add up to 180°… … the angles of the pentagon must add up to… 105° A … 180° x 3 = 540°… 105° B … the four given angles add up to 410°… … REDC = 130° © T Madas

A regular enneagon, (9 -sided polygon), is drawn below. Calculate the angles marked as x and y. 360° ÷ 9 = 40° x 70° 40° 70° y 180° – 40° = 140° ÷ 2 = 70° x = 140° y = 40° © T Madas

Calculate the exterior angle of a regular pentagon 360° ÷ 5 = 72° 180° – 72° = 108° ÷ 2 = 54° 72° 54° 54° 72° © T Madas

The diagram below shows part of a tessellation consisting of a regular octagon and a square. Show by a calculation that a regular hexagon cannot be used to fit the space between the square and the octagon. What regular polygon can fit instead? 360° ÷ 6 = 60° 180° – 60° = 120° 5° 13 135° 120° ÷ 2 = 60° 120° 60° 135° + 90° = 225° 360° – 225° = 135° 45° 5°. 7 6 The angle between the square and the octagon is 135° while the interior angle of a regular hexagon is 120°. Therefore a regular hexagon will not fit. © T Madas

The diagram below shows part of a tessellation consisting of a regular octagon and a square. Show by a calculation that a regular hexagon cannot be used to fit the space between the square and the octagon. What regular polygon can fit instead? 360° ÷ 6 = 60° 180° – 60° = 120° ÷ 2 = 60° 120° 5° 13 60° 135° + 90° = 225° 360° – 225° = 135° 45° 5°. 7 6 The angle between the square and the octagon is 135° while the interior angle of a regular hexagon is 120°. Therefore a regular hexagon will not fit. © T Madas

The diagram below shows part of a tessellation consisting of a regular octagon and a square. Show by a calculation that a regular hexagon cannot be used to fit the space between the square and the octagon. What regular polygon can fit instead? 360° ÷ 6 = 60° 180° – 60° = 120° ÷ 2 = 60° 120° 5° 13 60 ° 135° + 90° = 225° 360° – 225° = 135° 45° 5°. 7 6 The angle between the square and the octagon is 135° while the interior angle of a regular hexagon is 120°. Therefore a regular hexagon will not fit. © T Madas

5° 13 135° The diagram below shows part of a tessellation consisting of a regular octagon and a square. Show by a calculation that a regular hexagon cannot be used to fit the space between the square and the octagon. What regular polygon can fit instead? 45° 5°. 7 6 © T Madas

A tessellation of squares and regular octagons is shown opposite. Calculate the angle marked as x. x x angles around a point add up to … … 360° subtracting the right angle gives … … 270° dividing by 2 gives … 135° x = 135° © T Madas

Regular dodecagons tessellate with equilateral triangles as shown opposite. Use this fact to calculate the interior angle of a regular dodecagon. angles around a point add up to … … 360° subtracting the 60° angle gives … … 300° 60° x x dividing by 2 gives … 150° The interior angle of a regular dodecagon is 150° © T Madas

The sides of a regular pentagon are extended until they form the star shape shown below. Calculate the size of the angles marked as α and β. φ φ c c c α The central angle of a regular pentagon is given by: θ β 360° ÷ 5 = 72° θ = 72° φ = 54° β = 108° © T Madas

The sides of a regular pentagon are extended until they form the star shape shown below. Calculate the size of the angles marked as α and β. θ φ φ β θ c c α The central angle of a regular pentagon is given by: 360° ÷ 5 = 72° θ = 72° φ = 54° β = 108° α = 36° © T Madas

Calculate the area of a regular hexagon of side 8 cm, giving your answer to 3 significant figures. E D 4 30° h O F A 8 cm h h tan 30° = 4 4 h= tan 30° C 30° 4 cm 6. 928 60° = tan 30° h ≈ 6. 928 cm B © T Madas

Calculate the area of a regular hexagon of side 8 cm, giving your answer to 3 significant figures. E D O F C 6. 928 30° A 8 cm B © T Madas

Calculate the area of a regular hexagon of side 8 cm, giving your answer to 3 significant figures. E F 8 60° C cm A 60° 8 cm 60° AT = 32 sin 60° AH = 192 sin 60° AH = 166 cm 2 [ 3 s. f. ] x 8 x sin 60° c = c O AT c D B © T Madas

A regular enneagon (9 -sided polygon) ABCDEFGHI is shown below. Calculate RICE. I A ● for the circle circumscribing H ● ● ● B G O C ● the enneagon: RIOE is a central angle RICE is an inscribed angle both angles correspond to the same arc. hence RICE is half of RIOE ● The central angle of a regular F D E enneagon is given by: 360° ÷ 9 = 40° ● RIOE = 160° ● hence RICE = 80° © T Madas

18 identical slabs in the shape of isosceles trapeziums fit tightly around a small fish pond as shown below. What is the smallest and what is the largest angle of these trapezoidal slabs? 360° ÷ 18 = 20° © T Madas

A regular octagon is inscribed in a circle of radius 5 cm. Calculate the perimeter of the octagon, giving your answer correct to 3 significant figures. [AB ]2 = 52 + 52 – 2 x 5 x cos 45° A [AB ]2 = 25 + 25 – 50 cos 45° cm O [AB ]2 ≈ 14. 6467 5 45° B AB c c c By the cosine rule on OAB ≈ 3. 8268 cm The perimeter of the octagon 8 x 3. 8268 = 30. 6 cm [ 3 s. f. ] © T Madas

A regular decagon is inscribed in a circle of radius 4 cm. Calculate the area of the decagon, giving your answer correct to 3 significant figures. A = 1 2 A O 4 x sin 36° A ≈ 4. 702 cm 4 36° x c The area of the triangle OAB B The area of the decagon 10 x 4. 702 = 47. 0 cm 2 [ 3 s. f. ] © T Madas

A regular hexagon and a regular pentagon are made to overlap each other as shown in the diagram below. What is the size of the angle marked as x ? x © T Madas

A regular hexagon and a regular pentagon are made to overlap each other as shown in the diagram below. What is the size of the angle marked as x ? x How many degrees interior angle of a hexagon? is the regular … six equilateral triangles 120° 60° © T Madas

A regular hexagon and a regular pentagon are made to overlap each other as shown in the diagram below. What is the size of the angle marked as x ? © T Madas

A regular hexagon and a regular pentagon are made to overlap each other as shown in the diagram below. What is the size of the angle marked as x ? How many degrees interior angle of a pentagon? 72° 108° 54° is the regular … five isosceles triangles … 360° ÷ 5 = … … 72° … … 180° – 72° = … … 108° ÷ 2 = … … 54° … © T Madas

A regular hexagon and a regular pentagon are made to overlap each other as shown in the diagram below. What is the size of the angle marked as x ? x … split the overlapping pentagon into 3 triangles … … the angles of a triangle add up to 180°… 120° 108° … the angles of the pentagon must add up to… … 180° x 3 = 540°… … the four angles of the pentagon add up to 456°… … x = 84° © T Madas