AMC 8 Challenge Lab
Lesson 6: Geometry: Angles, Triangles & Polygons
Recognize • Model • Solve • Verify
50 original AMC 8-style multiple-choice problems with complete English solutions.
AMC 8 Strategy
Build speed without sacrificing accuracy.
1. Read for the exact target and hidden restrictions.
2. Translate words into a diagram, equation, table, or smaller case.
3. Estimate before calculating; eliminate impossible choices.
4. Look for shortcuts: symmetry, factoring, parity, ratios, or patterns.
5. Verify the result and make sure it answers the question asked.
Lesson Focus
Geometry: Angles, Triangles & Polygons
• Concept recognition
• Efficient contest strategy
• Mental-math checks
• Multiple representations
• Answer-choice verification
Practice 1-2
Choose the best answer. Solutions begin after Question 50.
1. What is the sum of the interior angles of a 5-gon?��A. 545 B. 720 C. 900 D. 360 E. 540
2. A right triangle has legs 7 and 24. What is its hypotenuse?��A. 25 B. 26 C. 31 D. 17 E. 84
Practice 3-4
Choose the best answer. Solutions begin after Question 50.
3. A rectangle is 12 units by 5 units. What is its area?��A. 17 B. 34 C. 60 D. 55 E. 72
4. What is the Manhattan distance from (1,1) to (5,3)?��A. 4 B. 6 C. 8 D. 2 E. 11
Practice 5-6
Choose the best answer. Solutions begin after Question 50.
5. If 6x+7=61, what is x?��A. 10 B. 8 C. 13 D. 9 E. 14
6. What is the sum of the interior angles of a 10-gon?��A. 1620 B. 1260 C. 1440 D. 1445 E. 1800
Practice 7-8
Choose the best answer. Solutions begin after Question 50.
7. A right triangle has legs 3 and 4. What is its hypotenuse?��A. 1 B. 10 C. 6 D. 5 E. 7
8. A rectangle is 17 units by 7 units. What is its area?��A. 48 B. 112 C. 24 D. 119 E. 136
Practice 9-10
Choose the best answer. Solutions begin after Question 50.
9. What is the Manhattan distance from (2,1) to (7,3)?��A. 2 B. 12 C. 5 D. 7 E. 9
10. If 5x+4=24, what is x?��A. 5 B. 4 C. 9 D. 3 E. 9
Practice 11-12
Choose the best answer. Solutions begin after Question 50.
11. What is the sum of the interior angles of a 9-gon?��A. 1440 B. 1260 C. 1620 D. 1265 E. 1080
12. A right triangle has legs 5 and 12. What is its hypotenuse?��A. 14 B. 7 C. 30 D. 17 E. 13
Practice 13-14
Choose the best answer. Solutions begin after Question 50.
13. A rectangle is 12 units by 9 units. What is its area?��A. 42 B. 108 C. 120 D. 21 E. 99
14. What is the Manhattan distance from (3,1) to (9,3)?��A. 2 B. 10 C. 8 D. 6 E. 13
Practice 15-16
Choose the best answer. Solutions begin after Question 50.
15. If 4x+1=37, what is x?��A. 14 B. 9 C. 8 D. 5 E. 10
16. What is the sum of the interior angles of a 8-gon?��A. 1085 B. 900 C. 1080 D. 1260 E. 1440
Practice 17-18
Choose the best answer. Solutions begin after Question 50.
17. A right triangle has legs 8 and 15. What is its hypotenuse?��A. 7 B. 18 C. 23 D. 60 E. 17
18. A rectangle is 17 units by 11 units. What is its area?��A. 28 B. 204 C. 176 D. 187 E. 56
Practice 19-20
Choose the best answer. Solutions begin after Question 50.
19. What is the Manhattan distance from (0,1) to (3,3)?��A. 7 B. 5 C. 10 D. 3 E. 2
20. If 3x+6=18, what is x?��A. 5 B. 6 C. 9 D. 3 E. 4
Practice 21-22
Choose the best answer. Solutions begin after Question 50.
21. What is the sum of the interior angles of a 7-gon?��A. 900 B. 720 C. 1260 D. 905 E. 1080
22. A right triangle has legs 7 and 24. What is its hypotenuse?��A. 26 B. 31 C. 17 D. 84 E. 25
Practice 23-24
Choose the best answer. Solutions begin after Question 50.
23. A rectangle is 12 units by 5 units. What is its area?��A. 34 B. 60 C. 72 D. 17 E. 55
24. What is the Manhattan distance from (1,1) to (5,3)?��A. 8 B. 6 C. 4 D. 2 E. 11
Practice 25-26
Choose the best answer. Solutions begin after Question 50.
25. If 2x+3=21, what is x?��A. 14 B. 10 C. 5 D. 9 E. 8
26. What is the sum of the interior angles of a 6-gon?��A. 720 B. 1080 C. 900 D. 725 E. 540
Practice 27-28
Choose the best answer. Solutions begin after Question 50.
27. A right triangle has legs 3 and 4. What is its hypotenuse?��A. 7 B. 5 C. 10 D. 6 E. 1
28. A rectangle is 17 units by 7 units. What is its area?��A. 119 B. 24 C. 112 D. 48 E. 136
Practice 29-30
Choose the best answer. Solutions begin after Question 50.
29. What is the Manhattan distance from (2,1) to (7,3)?��A. 5 B. 9 C. 7 D. 2 E. 12
30. If 7x+8=36, what is x?��A. 3 B. 15 C. 9 D. 5 E. 4
Practice 31-32
Choose the best answer. Solutions begin after Question 50.
31. What is the sum of the interior angles of a 5-gon?��A. 720 B. 540 C. 545 D. 360 E. 900
32. A right triangle has legs 5 and 12. What is its hypotenuse?��A. 17 B. 7 C. 14 D. 30 E. 13
Practice 33-34
Choose the best answer. Solutions begin after Question 50.
33. A rectangle is 12 units by 9 units. What is its area?��A. 108 B. 21 C. 42 D. 120 E. 99
34. What is the Manhattan distance from (3,1) to (9,3)?��A. 6 B. 10 C. 8 D. 13 E. 2
Practice 35-36
Choose the best answer. Solutions begin after Question 50.
35. If 6x+5=59, what is x?��A. 8 B. 11 C. 9 D. 14 E. 10
36. What is the sum of the interior angles of a 10-gon?��A. 1800 B. 1260 C. 1440 D. 1445 E. 1620
Practice 37-38
Choose the best answer. Solutions begin after Question 50.
37. A right triangle has legs 8 and 15. What is its hypotenuse?��A. 7 B. 60 C. 17 D. 23 E. 18
38. A rectangle is 17 units by 11 units. What is its area?��A. 28 B. 56 C. 176 D. 187 E. 204
Practice 39-40
Choose the best answer. Solutions begin after Question 50.
39. What is the Manhattan distance from (0,1) to (3,3)?��A. 7 B. 3 C. 5 D. 2 E. 10
40. If 5x+2=22, what is x?��A. 5 B. 4 C. 3 D. 7 E. 9
Practice 41-42
Choose the best answer. Solutions begin after Question 50.
41. What is the sum of the interior angles of a 9-gon?��A. 1080 B. 1260 C. 1620 D. 1265 E. 1440
42. A right triangle has legs 7 and 24. What is its hypotenuse?��A. 84 B. 26 C. 17 D. 31 E. 25
Practice 43-44
Choose the best answer. Solutions begin after Question 50.
43. A rectangle is 12 units by 5 units. What is its area?��A. 17 B. 60 C. 72 D. 55 E. 34
44. What is the Manhattan distance from (1,1) to (5,3)?��A. 4 B. 11 C. 6 D. 2 E. 8
Practice 45-46
Choose the best answer. Solutions begin after Question 50.
45. If 4x+7=43, what is x?��A. 8 B. 10 C. 11 D. 14 E. 9
46. What is the sum of the interior angles of a 8-gon?��A. 900 B. 1440 C. 1080 D. 1085 E. 1260
Practice 47-48
Choose the best answer. Solutions begin after Question 50.
47. A right triangle has legs 3 and 4. What is its hypotenuse?��A. 6 B. 10 C. 7 D. 5 E. 1
48. A rectangle is 17 units by 7 units. What is its area?��A. 112 B. 48 C. 24 D. 136 E. 119
Practice 49-50
Choose the best answer. Solutions begin after Question 50.
49. What is the Manhattan distance from (2,1) to (7,3)?��A. 5 B. 2 C. 7 D. 9 E. 12
50. If 3x+4=16, what is x?��A. 4 B. 7 C. 3 D. 5 E. 9
Answer Key
Questions 1-50
1. E�2. A�3. C�4. B�5. D�6. C�7. D�8. D�9. D�10. B
11. B�12. E�13. B�14. C�15. B�16. C�17. E�18. D�19. B�20. E
21. A�22. E�23. B�24. B�25. D�26. A�27. B�28. A�29. C�30. E
31. B�32. E�33. A�34. C�35. C�36. C�37. C�38. D�39. C�40. B
41. B�42. E�43. B�44. C�45. E�46. C�47. D�48. E�49. C�50. A
Solutions 1-2
Check the method, not just the letter.
1. Answer: E. 540��Use (n−2)180°: (5−2)180°=540°.
2. Answer: A. 25��By the Pythagorean theorem, c=√(7²+24²)=25.
Solutions 3-4
Check the method, not just the letter.
3. Answer: C. 60��Area = 12×5=60 square units.
4. Answer: B. 6��Add horizontal and vertical changes: 4+2=6.
Solutions 5-6
Check the method, not just the letter.
5. Answer: D. 9��Subtract 7, then divide by 6: x=(61−7)/6=9.
6. Answer: C. 1440��Use (n−2)180°: (10−2)180°=1440°.
Solutions 7-8
Check the method, not just the letter.
7. Answer: D. 5��By the Pythagorean theorem, c=√(3²+4²)=5.
8. Answer: D. 119��Area = 17×7=119 square units.
Solutions 9-10
Check the method, not just the letter.
9. Answer: D. 7��Add horizontal and vertical changes: 5+2=7.
10. Answer: B. 4��Subtract 4, then divide by 5: x=(24−4)/5=4.
Solutions 11-12
Check the method, not just the letter.
11. Answer: B. 1260��Use (n−2)180°: (9−2)180°=1260°.
12. Answer: E. 13��By the Pythagorean theorem, c=√(5²+12²)=13.
Solutions 13-14
Check the method, not just the letter.
13. Answer: B. 108��Area = 12×9=108 square units.
14. Answer: C. 8��Add horizontal and vertical changes: 6+2=8.
Solutions 15-16
Check the method, not just the letter.
15. Answer: B. 9��Subtract 1, then divide by 4: x=(37−1)/4=9.
16. Answer: C. 1080��Use (n−2)180°: (8−2)180°=1080°.
Solutions 17-18
Check the method, not just the letter.
17. Answer: E. 17��By the Pythagorean theorem, c=√(8²+15²)=17.
18. Answer: D. 187��Area = 17×11=187 square units.
Solutions 19-20
Check the method, not just the letter.
19. Answer: B. 5��Add horizontal and vertical changes: 3+2=5.
20. Answer: E. 4��Subtract 6, then divide by 3: x=(18−6)/3=4.
Solutions 21-22
Check the method, not just the letter.
21. Answer: A. 900��Use (n−2)180°: (7−2)180°=900°.
22. Answer: E. 25��By the Pythagorean theorem, c=√(7²+24²)=25.
Solutions 23-24
Check the method, not just the letter.
23. Answer: B. 60��Area = 12×5=60 square units.
24. Answer: B. 6��Add horizontal and vertical changes: 4+2=6.
Solutions 25-26
Check the method, not just the letter.
25. Answer: D. 9��Subtract 3, then divide by 2: x=(21−3)/2=9.
26. Answer: A. 720��Use (n−2)180°: (6−2)180°=720°.
Solutions 27-28
Check the method, not just the letter.
27. Answer: B. 5��By the Pythagorean theorem, c=√(3²+4²)=5.
28. Answer: A. 119��Area = 17×7=119 square units.
Solutions 29-30
Check the method, not just the letter.
29. Answer: C. 7��Add horizontal and vertical changes: 5+2=7.
30. Answer: E. 4��Subtract 8, then divide by 7: x=(36−8)/7=4.
Solutions 31-32
Check the method, not just the letter.
31. Answer: B. 540��Use (n−2)180°: (5−2)180°=540°.
32. Answer: E. 13��By the Pythagorean theorem, c=√(5²+12²)=13.
Solutions 33-34
Check the method, not just the letter.
33. Answer: A. 108��Area = 12×9=108 square units.
34. Answer: C. 8��Add horizontal and vertical changes: 6+2=8.
Solutions 35-36
Check the method, not just the letter.
35. Answer: C. 9��Subtract 5, then divide by 6: x=(59−5)/6=9.
36. Answer: C. 1440��Use (n−2)180°: (10−2)180°=1440°.
Solutions 37-38
Check the method, not just the letter.
37. Answer: C. 17��By the Pythagorean theorem, c=√(8²+15²)=17.
38. Answer: D. 187��Area = 17×11=187 square units.
Solutions 39-40
Check the method, not just the letter.
39. Answer: C. 5��Add horizontal and vertical changes: 3+2=5.
40. Answer: B. 4��Subtract 2, then divide by 5: x=(22−2)/5=4.
Solutions 41-42
Check the method, not just the letter.
41. Answer: B. 1260��Use (n−2)180°: (9−2)180°=1260°.
42. Answer: E. 25��By the Pythagorean theorem, c=√(7²+24²)=25.
Solutions 43-44
Check the method, not just the letter.
43. Answer: B. 60��Area = 12×5=60 square units.
44. Answer: C. 6��Add horizontal and vertical changes: 4+2=6.
Solutions 45-46
Check the method, not just the letter.
45. Answer: E. 9��Subtract 7, then divide by 4: x=(43−7)/4=9.
46. Answer: C. 1080��Use (n−2)180°: (8−2)180°=1080°.
Solutions 47-48
Check the method, not just the letter.
47. Answer: D. 5��By the Pythagorean theorem, c=√(3²+4²)=5.
48. Answer: E. 119��Area = 17×7=119 square units.
Solutions 49-50
Check the method, not just the letter.
49. Answer: C. 7��Add horizontal and vertical changes: 5+2=7.
50. Answer: A. 4��Subtract 4, then divide by 3: x=(16−4)/3=4.