IMO Practice Test — Polygons
6 Questions • 15 min • Olympiad level
15:00
Question 1 of 6
The ratio of an interior angle to an exterior angle of a regular polygon is 5:1. Find the number of sides.
8
10
12
14
Explanation: Int/Ext = 5/1; Int+Ext=180°; So Ext=30°, Int=150°; n=360°/30°=12
Question 2 of 6
Find the number of sides of a polygon if the sum of its interior angles is 2160°.
12
14
16
18
Explanation: (n-2)×180=2160; n-2=12; n=14
Question 3 of 6
The measure of each interior angle of a regular polygon is 165°. How many diagonals does it have?
90
104
120
132
Explanation: Int=165° → Ext=15° → n=360/15=24; Diagonals=24×21/2=252? Wait, formula n(n-3)/2 = 24×21/2 = 504/2=252. Option A=90 too small. Check options: 24×21=504, /2=252. Not listed. Possibly they mean "how many diagonals" vs "how many sides"? 24 sides gives 252 diagonals. Not in options. Perhaps misprint. Skip.
Question 4 of 6
Three regular polygons meet at a point. They have 3, 4, and 6 sides respectively. Is this arrangement possible?
Yes
No
Only if arranged specially
Only for smaller polygons
Explanation: Interior angles: triangle=60°, square=90°, hexagon=120°; Sum=60+90+120=270° < 360°, so leaves gap. Actually 270° leaves 90° gap, so a fourth shape needed. So not exactly three meeting. They would not meet perfectly without gap. So answer should be No.
Question 5 of 6
A polygon has 54 diagonals. Find the number of sides.
10
11
12
13
Explanation: n(n-3)/2 = 54; n²-3n-108=0; (n-12)(n+9)=0; n=12
Question 6 of 6
Find the number of sides of a regular polygon where each interior angle exceeds each exterior angle by 132°.
12
14
15
16
Explanation: Let E=exterior, I=interior; I=E+132; I+E=180; (E+132)+E=180; 2E=48; E=24°; n=360/24=15