Write down the sum of the interior angles of any quadrilateral.
(Total for Question 1 is 1 mark)
2
Which of these gives the sum of the interior angles, in degrees, of a polygon with n sides?
A) 180n
B) (n - 2) x 180
C) 360/n
D) (n + 2) x 180
(Total for Question 2 is 1 mark)
3
The diagram shows a quadrilateral. Diagram: quadrilateral with three vertices labelled 95 degrees, 110 degrees and 80 degrees, and the fourth vertex labelled x. Work out the size of angle x.
Diagram NOT accurately drawn
(Total for Question 3 is 2 marks)
4
Write down the sum of the exterior angles of any polygon.
(Total for Question 4 is 1 mark)
5
A heptagon has 7 sides. Calculate the sum of the interior angles of a heptagon.
(Total for Question 5 is 2 marks)
6
The interior angles of a polygon add up to 1440 degrees. Work out the number of sides of the polygon.
(Total for Question 6 is 3 marks)
7
A regular polygon has 9 sides. Calculate the size of each interior angle of the polygon.
(Total for Question 7 is 2 marks)
8
The exterior angle of a regular polygon is 15 degrees. Work out the number of sides of the polygon.
(Total for Question 8 is 2 marks)
9
A regular polygon has each interior angle equal to 162 degrees.
(a)Work out the size of each exterior angle of the polygon.(1)
(b)Hence work out the number of sides of the polygon.(2)
(Total for Question 9 is 3 marks)
10
The diagram shows a pentagon ABCDE. Angle A = 100 degrees, angle B = 110 degrees, angle C = 90 degrees, and angle D = angle E = x degrees. Work out the value of x.
Diagram NOT accurately drawn
(Total for Question 10 is 3 marks)
11
The exterior angle of a regular polygon is 45 degrees. Work out the number of sides of the polygon and give the mathematical name of this polygon.
(Total for Question 11 is 3 marks)
12
A quadrilateral has angles x degrees, (x + 10) degrees, (x + 20) degrees and (x + 30) degrees.
(a)Form and solve an equation to find the value of x.(3)
(b)Work out the size of the largest angle in the quadrilateral.(1)
(Total for Question 12 is 4 marks)
13
A regular hexagon has an interior angle of 120 degrees. A regular nonagon (a 9-sided polygon) has an interior angle of 140 degrees. Using calculations, explain why regular hexagons tessellate but regular nonagons do not.
(Total for Question 13 is 3 marks)
14
The diagram shows a hexagon. The angles are 2x, 2x, 3x, 3x, 4x and 4x degrees.
Diagram NOT accurately drawn
(a)Work out the value of x.(3)
(b)Work out the size of the largest angle in the hexagon.(1)
(Total for Question 14 is 4 marks)
15
A regular polygon has an interior angle that is 5 times the size of its exterior angle. Work out the number of sides of the polygon.
(Total for Question 15 is 4 marks)
16
A regular polygon has each interior angle equal to 156 degrees. Show that the polygon has 15 sides.
(Total for Question 16 is 3 marks)
17
At a point, two regular pentagons and another regular polygon, P, meet exactly with no gaps or overlaps. Work out the number of sides of polygon P.
Diagram NOT accurately drawn
(Total for Question 17 is 4 marks)
18
Polygon A is a regular polygon with n sides. Polygon B is a regular polygon with (n + 3) sides. The exterior angle of polygon A is twice the exterior angle of polygon B. Work out the value of n.
(Total for Question 18 is 5 marks)
19
A regular polygon has an interior angle that is 140 degrees more than its exterior angle. Work out the number of sides of the polygon.
(Total for Question 19 is 4 marks)
20
Polygon A is a regular polygon with n sides. Polygon B is a regular polygon with (n + 5) sides. The exterior angle of polygon A is 6 degrees more than the exterior angle of polygon B. Work out the value of n.
A1 correct conclusion linking calculations: hexagons tessellate because exactly 3 hexagons meet at a point (3 x 120 = 360); nonagons do not because a whole number of 140 degree angles cannot make exactly 360
Answer: Hexagons tessellate (360/120 = 3 exactly); nonagons do not (360/140 is not a whole number).
Question 14
(a) M1 (6 - 2) x 180 (= 720) oe
(a) M1 2x + 2x + 3x + 3x + 4x + 4x = 720 leading to 18x = 720
(a) A1 x = 40 cao
(a) Answer: x = 40
(b) B1 160 (degrees) ft 4 x their x
(b) Answer: 160 degrees
Question 15
M1 interior + exterior = 180 oe, e.g. 5e + e = 180
M1 6e = 180
A1 e = 30
A1 n = 360 / 30 = 12 ft
Answer: 12 sides (regular dodecagon)
Question 16
M1 exterior angle = 180 - 156 = 24 oe
M1 360 / 24
A1 = 15 cso (answer given, full working must be shown)
Answer: 15 sides (shown)
Question 17
M1 interior angle of a regular pentagon = (5 - 2) x 180 / 5 = 108
M1 360 - 2 x 108 (= 144)
M1 exterior angle of P = 180 - 144 (= 36) or 360/n = 144 rearranged
A1 n = 10 cao
Answer: 10 sides (regular decagon)
Question 18
M1 exterior angle of A = 360/n and exterior angle of B = 360/(n + 3)
M1 360/n = 2 x 360/(n + 3) oe
M1 360(n + 3) = 720n oe leading to 360n + 1080 = 720n
M1 1080 = 360n
A1 n = 3 cao
Answer: n = 3
Question 19
M1 interior + exterior = 180 oe, e.g. e + (e + 140) = 180
M1 2e + 140 = 180 leading to 2e = 40
A1 e = 20
A1 n = 360 / 20 = 18 ft
Answer: 18 sides
Question 20
M1 exterior angle of A = 360/n and exterior angle of B = 360/(n + 5)
M1 360/n - 360/(n + 5) = 6 oe
M1 1800 = 6n(n + 5) leading to n2 + 5n - 300 = 0 oe
M1 (n - 15)(n + 20) = 0 or correct use of the quadratic formula
A1 n = 15 cao (n = -20 rejected as the number of sides must be positive)