State, in degrees, the sum of the interior angles of a triangle.
(Total for Question 1 is 1 mark)
2
A polygon has 13 sides. Calculate the sum of its interior angles.
(Total for Question 2 is 2 marks)
3
A polygon has 11 sides. Calculate the sum of its interior angles.
(Total for Question 3 is 2 marks)
4
A polygon has 14 sides. Calculate the sum of its interior angles.
(Total for Question 4 is 2 marks)
5
A regular polygon has 45 sides. Work out the size of each interior angle.
(Total for Question 5 is 2 marks)
6
A regular polygon has 30 sides. Work out the size of each exterior angle.
(Total for Question 6 is 2 marks)
7
The exterior angle of a regular polygon is 9 degrees. Work out the number of sides of the polygon.
(Total for Question 7 is 2 marks)
8
The interior angle of a regular polygon is 170 degrees. Work out the number of sides of the polygon.
(Total for Question 8 is 3 marks)
9
The interior angle of a regular polygon is 174 degrees. Work out the number of sides of the polygon.
(Total for Question 9 is 3 marks)
10
In a quadrilateral, three of the angles measure 97 degrees, 112 degrees and 94 degrees. Work out the size of the fourth angle.
(Total for Question 10 is 2 marks)
11
In a pentagon, four of the angles measure 118 degrees, 96 degrees, 102 degrees and 89 degrees. Work out the size of the fifth angle.
(Total for Question 11 is 2 marks)
12
A regular polygon has 72 sides. Work out the size of each exterior angle.
(Total for Question 12 is 2 marks)
13
A polygon has 16 sides. Calculate the sum of its interior angles.
(Total for Question 13 is 2 marks)
14
A polygon has 100 sides. State the sum of its exterior angles.
(Total for Question 14 is 1 mark)
15
For a certain regular polygon, the sum of its interior angles is 10 times the sum of its exterior angles. Work out the number of sides of the polygon.
(Total for Question 15 is 3 marks)
16
Nadia says: 'A regular polygon can have an interior angle of exactly 173 degrees.' Show, using calculations, that Nadia is wrong.
(Total for Question 16 is 3 marks)
17
In a pentagon, three of the angles measure 101 degrees, 109 degrees and 113 degrees. The remaining two angles are in the ratio 3 : 4.
(a)Form and solve an equation to find the value of one part of the ratio.(3)
(b)Hence work out the size of the larger of the two remaining angles.(1)
(Total for Question 17 is 4 marks)
18
An irregular pentagon has exterior angles of 58 degrees, 71 degrees, 91 degrees, 63 degrees and y degrees.
(a)Work out the value of y.(2)
(b)Work out the size of the interior angle at the vertex where the exterior angle is y degrees.(1)
(Total for Question 18 is 3 marks)
19
The six exterior angles of a hexagon, in degrees, are x, x + 4, x + 8, x + 12, x + 16 and x + 20.
(a)Show that 6x + 60 = 360.(1)
(b)Solve the equation to find the value of x.(2)
(c)Work out the size of the largest interior angle of the hexagon.(2)
(Total for Question 19 is 5 marks)
20
A dart-shaped (concave) quadrilateral has three interior angles of 43 degrees, 51 degrees and 38 degrees, and a reflex interior angle at the fourth vertex. Work out the size of the reflex angle, explaining how you know it is reflex.
(Total for Question 20 is 3 marks)
21
A polygon gains 3 extra sides (no vertex becomes a straight line). By how much does the sum of its interior angles increase?
(Total for Question 21 is 2 marks)
22
Priya is designing a tiled floor using two regular octagons and one square meeting at every vertex, with no gaps or overlaps. Show, using calculations, that this arrangement fits exactly around a point.
(Total for Question 22 is 3 marks)
23
Let sum(n) = (n - 2) * 180 be the sum of the interior angles of a polygon with n sides.
(a)Prove algebraically that sum(n + 1) - sum(n) = 180 for every whole number n, showing that increasing the number of sides of a polygon by 1 always increases the sum of its interior angles by exactly 180 degrees.(3)
(b)A polygon with 22 sides has an interior angle sum of 3600 degrees. Using the result from part (a), write down the sum of the interior angles of a polygon with 23 sides.(2)
(Total for Question 23 is 5 marks)
Mark scheme · 4.11D Angles in Polygons: Fluency and Exam Drill
Question 1
B1 180 (degrees) cao
Answer: 180 degrees
Question 2
M1 (13 - 2) * 180 oe
A1 1980 (degrees) cao
Answer: 1980 degrees
Question 3
M1 (11 - 2) * 180 oe
A1 1620 (degrees) cao
Answer: 1620 degrees
Question 4
M1 (14 - 2) * 180 oe
A1 2160 (degrees) cao
Answer: 2160 degrees
Question 5
M1 (45 - 2) * 180 / 45 oe, or (45 - 2) * 180 (= 7740) soi
A1 172 (degrees) cao
Answer: 172 degrees
Question 6
M1 360 / 30 soi
A1 12 (degrees) cao
Answer: 12 degrees
Question 7
M1 360 / 9 soi
A1 40 (sides) cao
Answer: 40 sides
Question 8
M1 180 - 170 (= 10) to find the exterior angle oe
M1 360 / their 10 (dependent on the previous method mark)
A1 36 (sides) cao
Answer: 36 sides
Question 9
M1 180 - 174 (= 6) to find the exterior angle oe
M1 360 / their 6 (dependent on the previous method mark)
A1 60 (sides) cao
Answer: 60 sides
Question 10
M1 97 + 112 + 94 (= 303) soi
A1 57 (degrees) cao
Answer: 57 degrees
Question 11
M1 (5 - 2) * 180 (= 540) and 118 + 96 + 102 + 89 (= 405) soi
A1 135 (degrees) cao
Answer: 135 degrees
Question 12
M1 360 / 72 soi
A1 5 (degrees) cao
Answer: 5 degrees
Question 13
M1 (16 - 2) * 180 oe
A1 2520 (degrees) cao
Answer: 2520 degrees
Question 14
B1 360 (degrees) cao
Answer: 360 degrees
Question 15
M1 10 * 360 (= 3600), using sum of exterior angles = 360
M1 (n - 2) * 180 = 3600 leading to n - 2 = 20
A1 n = 22 cao
Answer: 22 sides
Question 16
M1 180 - 173 (= 7) to find the exterior angle oe
M1 360 / their 7 (dependent on the previous method mark)
A1 awrt 51.4, which is not a whole number, with a conclusion that a regular polygon cannot have an interior angle of exactly 173 degrees
Answer: 360 / 7 = awrt 51.4, which is not a whole number, so Nadia is wrong.
Question 17
(a) M1 540 - (101 + 109 + 113) (= 217) oe
(a) M1 3y + 4y = 217 (or 7y = 217)
(a) A1 y = 31 cao
(a) Answer: 31 degrees
(b) B1 124 (degrees), ft 4 * their y from part (a)
(b) Answer: 124 degrees
Question 18
(a) M1 58 + 71 + 91 + 63 (= 283) soi, using sum of exterior angles = 360
(a) A1 y = 77 cao
(a) Answer: y = 77
(b) B1 103 (degrees), ft 180 - their y
(b) Answer: 103 degrees
Question 19
(a) B1 x + (x + 4) + (x + 8) + (x + 12) + (x + 16) + (x + 20) = 360 correctly collected to 6x + 60 = 360, using sum of exterior angles = 360
(a) Answer: 6x + 60 = 360 (shown)
(b) M1 6x = 300
(b) A1 x = 50 cao
(b) Answer: x = 50
(c) M1 identifies the smallest exterior angle as their x (= 50) and finds 180 - their x
(c) A1 130 (degrees) cao (ft their x)
(c) Answer: 130 degrees
Question 20
M1 43 + 51 + 38 (= 132) soi
M1 360 - their 132
A1 228 (degrees), with a correct explanation that 228 is greater than 180 degrees, confirming it is reflex
Answer: 228 degrees (reflex, since 228 > 180)
Question 21
M1 3 * 180 oe, using the fact that each extra side adds 180 degrees to the interior angle sum
A1 540 (degrees) cao
Answer: 540 degrees
Question 22
M1 interior angle of a regular octagon = (8 - 2) * 180 / 8 (= 135) soi
B1 interior angle of a square = 90 stated
A1 135 + 135 + 90 = 360, with a conclusion that the angles fit exactly around the point with no gap or overlap
Answer: 135 + 135 + 90 = 360, so two regular octagons and one square fit exactly around a point.