Find the sum of the interior angles of a regular 12-gon.
(Total for Question 2 is 3 marks)
3
Work out the exterior angle of a regular heptagon. Give your answer exactly and to 2 decimal places.
(Total for Question 3 is 3 marks)
4
A nonagon (9-sided polygon) has one interior angle equal to 150 degrees. The remaining eight interior angles are all equal. Work out the size of each of the eight equal angles.
(Total for Question 4 is 4 marks)
5
A regular polygon has interior angle 156 degrees. Find the number of sides of the polygon.
(Total for Question 5 is 4 marks)
6
Show that the interior angle of a regular n-sided polygon is 180 - 360/n degrees. You must justify each step.
(Total for Question 6 is 5 marks)
7
A convex 10-sided polygon has two interior angles equal to 120 degrees and 130 degrees. The other eight interior angles are all equal. Work out the size of each of the eight equal angles.
(Total for Question 7 is 5 marks)
8
Show that the sum of the exterior angles, taking one exterior angle at each vertex, of any convex polygon is 360 degrees. Give a clear justification.
(Total for Question 8 is 7 marks)
9
A regular 12-sided polygon is altered so that the 12 interior angles alternate between (3x + 10) degrees and (2x + 20) degrees around the polygon (so six angles are 3x + 10 and six angles are 2x + 20). Find x and then work out the sizes of both types of angle. State whether the resulting polygon is convex.
(Total for Question 9 is 7 marks)
Mark scheme · 4.11H Angles in Polygons: Higher Tier Practice
Question 1
M1 use (n - 2) * 180 / n for a regular n-gon with n = 8
Answer: 360/7 degrees, which is approximately 51.43 degrees (360/7 = 51.428571... so approximately 51.43 degrees).
Question 4
M1 sum of interior angles = (9 - 2) * 180 = 7 * 180
M1 subtract the given 150 degrees: 1260 - 150
M1 divide the remaining sum by 8 to find each equal angle
A1 138.75 degrees cao
Answer: Each of the eight equal angles is 138.75 degrees, since total = 7 * 180 = 1260; remaining = 1260 - 150 = 1110; each = 1110 / 8 = 138.75.
Question 5
M1 set (n - 2) * 180 / n = 156
M1 rearrange: 180n - 360 = 156n
A1 n = 15 cao
B1 verify n is an integer and > 2
Answer: 15 sides, since (n - 2) * 180 = 156n gives 180n - 360 = 156n so 24n = 360 and n = 15.
Question 6
dM1 state sum of interior angles = (n - 2) * 180
M1 divide sum by n to get each interior angle = (n - 2) * 180 / n
M1 expand/ rearrange (n - 2) * 180 / n = (180n - 360) / n
M1 simplify (180n - 360) / n to 180 - 360 / n
A1 final form 180 - 360/n cao and clear justification
Answer: Starting from sum = (n - 2) * 180, each interior angle = ((n - 2) * 180) / n = (180n - 360) / n = 180 - 360 / n, so interior angle = 180 - 360/n degrees. Exterior angle = 180 - interior = 360 / n, consistent with the result above.
Question 7
M1 use sum of interior angles = (10 - 2) * 180 = 1440
M1 subtract the two given angles: 1440 - (120 + 130)
M1 compute remaining sum = 1440 - 250 = 1190
M1 divide by 8 to get each angle
A1 148.75 degrees cao
Answer: Each equal angle is 148.75 degrees, since total 1440, remaining 1440 - 250 = 1190, so each = 1190 / 8 = 148.75.
Question 8
dM1 define exterior angle at a vertex as 180 - interior angle (or state interior + exterior = 180)
M1 write sum of interior angles = (n - 2) * 180
M1 write sum of n exterior angles = n * 180 - sum of interior angles
M1 substitute sum of interior angles to get n * 180 - (n - 2) * 180
M1 simplify to (n * 180 - n * 180 + 360) = 360
A1 conclude sum of exterior angles = 360 degrees cao
B1 clear statement that this holds for any convex polygon using one exterior at each vertex
Answer: 360 degrees. Using exterior = 180 - interior and sum interior = (n - 2) * 180, sum exterior = n*180 - (n - 2)*180 = 360, so the sum is 360 degrees for any convex polygon taking one exterior angle at each vertex.
Question 9
M1 set up equation: 6*(3x + 10) + 6*(2x + 20) = sum interior angles = (12 - 2) * 180
M1 simplify left side: 6*(5x + 30) = 30x + 180
M1 write equation 30x + 180 = 1800
M1 solve for x: 30x = 1620 so x = 54
M1 substitute x to find angles: 3x + 10 = 3*54 + 10 = 172, and 2x + 20 = 2*54 + 20 = 128
A1 state both angles: 172 degrees and 128 degrees cao
B1 state polygon is convex since all interior angles are less than 180 degrees
Answer: x = 54. The two angle sizes are 3x + 10 = 172 degrees and 2x + 20 = 128 degrees. The polygon is convex because all interior angles are less than 180 degrees.