MATHCOUNTS Chapter Lab

Lesson 6: Geometry I: Angles, Triangles & Polygons

Solve • Strategize • Check • Explain

Chapter-level practice with 50 multiple-choice problems and complete solutions.

MATHCOUNTS Chapter Lab — Lesson 6

Lesson Method

Use a contest-style checklist before choosing an answer.

1. Decode

Read for hidden constraints: integer, positive, inclusive, least, greatest.

2. Choose a Tool

Factorization, ratios, equations, diagrams, counting, or modular arithmetic.

3. Estimate

Use mental checks to eliminate unreasonable answers.

4. Solve Cleanly

Write only enough work to avoid mistakes under time pressure.

5. Verify

Check units, signs, parity, and whether the answer satisfies the question.

MATHCOUNTS Chapter Lab — Lesson 6

Topic Focus

The goal is fast recognition of structure, not long computation.

• Identify the mathematical structure before calculating.

• Use multiple approaches: arithmetic, algebra, diagrams and estimation.

• Keep choices hidden from your reasoning until you have a target answer.

• When stuck, test answer choices or use a smaller case.

MATHCOUNTS Chapter Lab — Lesson 6

Sprint Practice 1-2

Choose the best answer. Answers are not shown in this section.

1. What is the sum of the interior angles of a 10-sided polygon?��A. 1800�B. 1260�C. 1440�D. 1620

2. A triangle has base 12 and height 9. What is its area?��A. 54�B. 59�C. 108�D. 49

MATHCOUNTS Chapter Lab — Lesson 6

Sprint Practice 3-4

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3. A right triangle has legs 9 and 12. What is the hypotenuse?��A. 3�B. 15�C. 16�D. 21

4. A rectangle has length 12 and width 11. What is its area?��A. 23�B. 132�C. 144�D. 46

MATHCOUNTS Chapter Lab — Lesson 6

Sprint Practice 5-6

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5. A circle has radius 3. Its area is kπ. What is k?��A. 15�B. 3�C. 6�D. 9

6. A rectangular prism has dimensions 5, 6 and 6. What is its volume?��A. 17�B. 192�C. 186�D. 180

MATHCOUNTS Chapter Lab — Lesson 6

Sprint Practice 7-8

Choose the best answer. Answers are not shown in this section.

7. What is the Manhattan distance between (3,4) and (9,10)?��A. 8�B. 14�C. 12�D. 6

8. What is the slope of the line through (1,6) and (3,16)?��A. 2�B. 5�C. 6�D. 4

MATHCOUNTS Chapter Lab — Lesson 6

Sprint Practice 9-10

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9. A ratio is 2:4. If the total is 30, what is the smaller/larger first part?��A. 6�B. 10�C. 20�D. 12

10. What is 20% of 160?��A. 32�B. 37�C. 800�D. 27

MATHCOUNTS Chapter Lab — Lesson 6

Sprint Practice 11-12

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11. What is the sum of the interior angles of a 8-sided polygon?��A. 1080�B. 900�C. 1260�D. 1440

12. A triangle has base 16 and height 5. What is its area?��A. 35�B. 80�C. 40�D. 45

MATHCOUNTS Chapter Lab — Lesson 6

Sprint Practice 13-14

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13. A right triangle has legs 9 and 12. What is the hypotenuse?��A. 21�B. 15�C. 3�D. 16

14. A rectangle has length 13 and width 10. What is its area?��A. 23�B. 143�C. 130�D. 46

MATHCOUNTS Chapter Lab — Lesson 6

Sprint Practice 15-16

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15. A circle has radius 6. Its area is kπ. What is k?��A. 6�B. 36�C. 12�D. 48

16. A rectangular prism has dimensions 5, 4 and 7. What is its volume?��A. 16�B. 140�C. 147�D. 166

MATHCOUNTS Chapter Lab — Lesson 6

Sprint Practice 17-18

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17. What is the Manhattan distance between (3,2) and (9,6)?��A. 12�B. 4�C. 6�D. 10

18. What is the slope of the line through (3,6) and (7,26)?��A. 5�B. 4�C. 6�D. 20

MATHCOUNTS Chapter Lab — Lesson 6

Sprint Practice 19-20

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19. A ratio is 2:6. If the total is 40, what is the smaller/larger first part?��A. 10�B. 30�C. 4�D. 12

20. What is 20% of 80?��A. 400�B. 16�C. 21�D. 11

MATHCOUNTS Chapter Lab — Lesson 6

Sprint Practice 21-22

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21. What is the sum of the interior angles of a 6-sided polygon?��A. 1080�B. 540�C. 900�D. 720

22. A triangle has base 20 and height 8. What is its area?��A. 85�B. 80�C. 160�D. 75

MATHCOUNTS Chapter Lab — Lesson 6

Sprint Practice 23-24

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23. A right triangle has legs 9 and 12. What is the hypotenuse?��A. 16�B. 15�C. 21�D. 3

24. A rectangle has length 14 and width 9. What is its area?��A. 23�B. 46�C. 126�D. 140

MATHCOUNTS Chapter Lab — Lesson 6

Sprint Practice 25-26

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25. A circle has radius 9. Its area is kπ. What is k?��A. 18�B. 99�C. 9�D. 81

26. A rectangular prism has dimensions 5, 6 and 5. What is its volume?��A. 155�B. 150�C. 170�D. 16

MATHCOUNTS Chapter Lab — Lesson 6

Sprint Practice 27-28

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27. What is the Manhattan distance between (3,4) and (9,9)?��A. 5�B. 6�C. 13�D. 11

28. What is the slope of the line through (1,6) and (3,16)?��A. 6�B. 2�C. 4�D. 5

MATHCOUNTS Chapter Lab — Lesson 6

Sprint Practice 29-30

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29. A ratio is 2:4. If the total is 30, what is the smaller/larger first part?��A. 20�B. 12�C. 6�D. 10

30. What is 20% of 90?��A. 450�B. 18�C. 13�D. 23

MATHCOUNTS Chapter Lab — Lesson 6

Sprint Practice 31-32

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31. What is the sum of the interior angles of a 10-sided polygon?��A. 1620�B. 1440�C. 1260�D. 1800

32. A triangle has base 8 and height 11. What is its area?��A. 88�B. 49�C. 44�D. 39

MATHCOUNTS Chapter Lab — Lesson 6

Sprint Practice 33-34

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33. A right triangle has legs 9 and 12. What is the hypotenuse?��A. 3�B. 16�C. 15�D. 21

34. A rectangle has length 15 and width 8. What is its area?��A. 23�B. 135�C. 120�D. 46

MATHCOUNTS Chapter Lab — Lesson 6

Sprint Practice 35-36

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35. A circle has radius 5. Its area is kπ. What is k?��A. 5�B. 10�C. 35�D. 25

36. A rectangular prism has dimensions 5, 4 and 6. What is its volume?��A. 148�B. 15�C. 120�D. 126

MATHCOUNTS Chapter Lab — Lesson 6

Sprint Practice 37-38

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37. What is the Manhattan distance between (3,2) and (9,8)?��A. 14�B. 8�C. 6�D. 12

38. What is the slope of the line through (3,6) and (7,26)?��A. 4�B. 20�C. 6�D. 5

MATHCOUNTS Chapter Lab — Lesson 6

Sprint Practice 39-40

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39. A ratio is 2:6. If the total is 40, what is the smaller/larger first part?��A. 12�B. 4�C. 30�D. 10

40. What is 20% of 100?��A. 25�B. 20�C. 500�D. 15

MATHCOUNTS Chapter Lab — Lesson 6

Sprint Practice 41-42

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41. What is the sum of the interior angles of a 8-sided polygon?��A. 900�B. 1080�C. 1440�D. 1260

42. A triangle has base 12 and height 7. What is its area?��A. 42�B. 47�C. 37�D. 84

MATHCOUNTS Chapter Lab — Lesson 6

Sprint Practice 43-44

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43. A right triangle has legs 9 and 12. What is the hypotenuse?��A. 21�B. 16�C. 3�D. 15

44. A rectangle has length 16 and width 7. What is its area?��A. 23�B. 46�C. 128�D. 112

MATHCOUNTS Chapter Lab — Lesson 6

Sprint Practice 45-46

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45. A circle has radius 8. Its area is kπ. What is k?��A. 64�B. 16�C. 8�D. 80

46. A rectangular prism has dimensions 5, 6 and 7. What is its volume?��A. 18�B. 210�C. 214�D. 217

MATHCOUNTS Chapter Lab — Lesson 6

Sprint Practice 47-48

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47. What is the Manhattan distance between (3,4) and (9,8)?��A. 6�B. 4�C. 10�D. 12

48. What is the slope of the line through (1,6) and (3,16)?��A. 2�B. 6�C. 5�D. 4

MATHCOUNTS Chapter Lab — Lesson 6

Sprint Practice 49-50

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49. A ratio is 2:4. If the total is 30, what is the smaller/larger first part?��A. 6�B. 10�C. 20�D. 12

50. What is 20% of 110?��A. 22�B. 17�C. 27�D. 550

MATHCOUNTS Chapter Lab — Lesson 6

Answer Key

Multiple-choice questions 1-50

1. C�2. A�3. B�4. B�5. D�6. D�7. C�8. B�9. B�10. A

11. A�12. C�13. B�14. C�15. B�16. B�17. D�18. A�19. A�20. B

21. D�22. B�23. B�24. C�25. D�26. B�27. D�28. D�29. D�30. B

31. B�32. C�33. C�34. C�35. D�36. C�37. D�38. D�39. D�40. B

41. B�42. A�43. D�44. D�45. A�46. B�47. C�48. C�49. B�50. A

MATHCOUNTS Chapter Lab — Lesson 6

Solutions 1-2

Read after completing the practice section.

1. Answer: C. 1440�The sum of interior angles of an n-gon is (n-2)180. For n=10, the sum is (10-2)180 = 1440.

2. Answer: A. 54�Triangle area = (1/2)bh = (1/2)(12)(9) = 54.

MATHCOUNTS Chapter Lab — Lesson 6

Solutions 3-4

Read after completing the practice section.

3. Answer: B. 15�By the Pythagorean theorem, c^2 = 9^2+12^2 = 225, so c = 15.

4. Answer: B. 132�Area of a rectangle is length × width = 12 × 11 = 132.

MATHCOUNTS Chapter Lab — Lesson 6

Solutions 5-6

Read after completing the practice section.

5. Answer: D. 9�Circle area = πr^2. With radius 3, area = 9π, so k = 9.

6. Answer: D. 180�Volume = length × width × height = 5×6×6 = 180.

MATHCOUNTS Chapter Lab — Lesson 6

Solutions 7-8

Read after completing the practice section.

7. Answer: C. 12�Manhattan distance is horizontal change plus vertical change: |9-3| + |10-4| = 12.

8. Answer: B. 5�Slope = change in y / change in x = (16-6)/(3-1) = 10/2 = 5.

MATHCOUNTS Chapter Lab — Lesson 6

Solutions 9-10

Read after completing the practice section.

9. Answer: B. 10�The total ratio parts are 6. One part is 30/6 = 5. The first part is 2 parts, or 10.

10. Answer: A. 32�20% = 20/100. So 20% of 160 is 20/100 × 160 = 32.

MATHCOUNTS Chapter Lab — Lesson 6

Solutions 11-12

Read after completing the practice section.

11. Answer: A. 1080�The sum of interior angles of an n-gon is (n-2)180. For n=8, the sum is (8-2)180 = 1080.

12. Answer: C. 40�Triangle area = (1/2)bh = (1/2)(16)(5) = 40.

MATHCOUNTS Chapter Lab — Lesson 6

Solutions 13-14

Read after completing the practice section.

13. Answer: B. 15�By the Pythagorean theorem, c^2 = 9^2+12^2 = 225, so c = 15.

14. Answer: C. 130�Area of a rectangle is length × width = 13 × 10 = 130.

MATHCOUNTS Chapter Lab — Lesson 6

Solutions 15-16

Read after completing the practice section.

15. Answer: B. 36�Circle area = πr^2. With radius 6, area = 36π, so k = 36.

16. Answer: B. 140�Volume = length × width × height = 5×4×7 = 140.

MATHCOUNTS Chapter Lab — Lesson 6

Solutions 17-18

Read after completing the practice section.

17. Answer: D. 10�Manhattan distance is horizontal change plus vertical change: |9-3| + |6-2| = 10.

18. Answer: A. 5�Slope = change in y / change in x = (26-6)/(7-3) = 20/4 = 5.

MATHCOUNTS Chapter Lab — Lesson 6

Solutions 19-20

Read after completing the practice section.

19. Answer: A. 10�The total ratio parts are 8. One part is 40/8 = 5. The first part is 2 parts, or 10.

20. Answer: B. 16�20% = 20/100. So 20% of 80 is 20/100 × 80 = 16.

MATHCOUNTS Chapter Lab — Lesson 6

Solutions 21-22

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21. Answer: D. 720�The sum of interior angles of an n-gon is (n-2)180. For n=6, the sum is (6-2)180 = 720.

22. Answer: B. 80�Triangle area = (1/2)bh = (1/2)(20)(8) = 80.

MATHCOUNTS Chapter Lab — Lesson 6

Solutions 23-24

Read after completing the practice section.

23. Answer: B. 15�By the Pythagorean theorem, c^2 = 9^2+12^2 = 225, so c = 15.

24. Answer: C. 126�Area of a rectangle is length × width = 14 × 9 = 126.

MATHCOUNTS Chapter Lab — Lesson 6

Solutions 25-26

Read after completing the practice section.

25. Answer: D. 81�Circle area = πr^2. With radius 9, area = 81π, so k = 81.

26. Answer: B. 150�Volume = length × width × height = 5×6×5 = 150.

MATHCOUNTS Chapter Lab — Lesson 6

Solutions 27-28

Read after completing the practice section.

27. Answer: D. 11�Manhattan distance is horizontal change plus vertical change: |9-3| + |9-4| = 11.

28. Answer: D. 5�Slope = change in y / change in x = (16-6)/(3-1) = 10/2 = 5.

MATHCOUNTS Chapter Lab — Lesson 6

Solutions 29-30

Read after completing the practice section.

29. Answer: D. 10�The total ratio parts are 6. One part is 30/6 = 5. The first part is 2 parts, or 10.

30. Answer: B. 18�20% = 20/100. So 20% of 90 is 20/100 × 90 = 18.

MATHCOUNTS Chapter Lab — Lesson 6

Solutions 31-32

Read after completing the practice section.

31. Answer: B. 1440�The sum of interior angles of an n-gon is (n-2)180. For n=10, the sum is (10-2)180 = 1440.

32. Answer: C. 44�Triangle area = (1/2)bh = (1/2)(8)(11) = 44.

MATHCOUNTS Chapter Lab — Lesson 6

Solutions 33-34

Read after completing the practice section.

33. Answer: C. 15�By the Pythagorean theorem, c^2 = 9^2+12^2 = 225, so c = 15.

34. Answer: C. 120�Area of a rectangle is length × width = 15 × 8 = 120.

MATHCOUNTS Chapter Lab — Lesson 6

Solutions 35-36

Read after completing the practice section.

35. Answer: D. 25�Circle area = πr^2. With radius 5, area = 25π, so k = 25.

36. Answer: C. 120�Volume = length × width × height = 5×4×6 = 120.

MATHCOUNTS Chapter Lab — Lesson 6

Solutions 37-38

Read after completing the practice section.

37. Answer: D. 12�Manhattan distance is horizontal change plus vertical change: |9-3| + |8-2| = 12.

38. Answer: D. 5�Slope = change in y / change in x = (26-6)/(7-3) = 20/4 = 5.

MATHCOUNTS Chapter Lab — Lesson 6

Solutions 39-40

Read after completing the practice section.

39. Answer: D. 10�The total ratio parts are 8. One part is 40/8 = 5. The first part is 2 parts, or 10.

40. Answer: B. 20�20% = 20/100. So 20% of 100 is 20/100 × 100 = 20.

MATHCOUNTS Chapter Lab — Lesson 6

Solutions 41-42

Read after completing the practice section.

41. Answer: B. 1080�The sum of interior angles of an n-gon is (n-2)180. For n=8, the sum is (8-2)180 = 1080.

42. Answer: A. 42�Triangle area = (1/2)bh = (1/2)(12)(7) = 42.

MATHCOUNTS Chapter Lab — Lesson 6

Solutions 43-44

Read after completing the practice section.

43. Answer: D. 15�By the Pythagorean theorem, c^2 = 9^2+12^2 = 225, so c = 15.

44. Answer: D. 112�Area of a rectangle is length × width = 16 × 7 = 112.

MATHCOUNTS Chapter Lab — Lesson 6

Solutions 45-46

Read after completing the practice section.

45. Answer: A. 64�Circle area = πr^2. With radius 8, area = 64π, so k = 64.

46. Answer: B. 210�Volume = length × width × height = 5×6×7 = 210.

MATHCOUNTS Chapter Lab — Lesson 6

Solutions 47-48

Read after completing the practice section.

47. Answer: C. 10�Manhattan distance is horizontal change plus vertical change: |9-3| + |8-4| = 10.

48. Answer: C. 5�Slope = change in y / change in x = (16-6)/(3-1) = 10/2 = 5.

MATHCOUNTS Chapter Lab — Lesson 6

Solutions 49-50

Read after completing the practice section.

49. Answer: B. 10�The total ratio parts are 6. One part is 30/6 = 5. The first part is 2 parts, or 10.

50. Answer: A. 22�20% = 20/100. So 20% of 110 is 20/100 × 110 = 22.

MATHCOUNTS Chapter Lab — Lesson 6