MATHCOUNTS Chapter Lab
Lesson 7: Geometry II: Circles, Area, Surface & Volume
Solve • Strategize • Check • Explain
Chapter-level practice with 50 multiple-choice problems and complete solutions.
MATHCOUNTS Chapter Lab — Lesson 7
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 7
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 7
Sprint Practice 1-2
Choose the best answer. Answers are not shown in this section.
1. A circle has radius 8. Its area is kπ. What is k?��A. 80�B. 16�C. 64�D. 8
2. A rectangle has length 11 and width 11. What is its area?��A. 132�B. 121�C. 44�D. 22
MATHCOUNTS Chapter Lab — Lesson 7
Sprint Practice 3-4
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3. A triangle has base 8 and height 5. What is its area?��A. 20�B. 40�C. 15�D. 25
4. A rectangular prism has dimensions 5, 5 and 5. What is its volume?��A. 130�B. 150�C. 125�D. 15
MATHCOUNTS Chapter Lab — Lesson 7
Sprint Practice 5-6
Choose the best answer. Answers are not shown in this section.
5. A right triangle has legs 7 and 24. What is the hypotenuse?��A. 26�B. 31�C. 17�D. 25
6. What is the sum of the interior angles of a 10-sided polygon?��A. 1260�B. 1440�C. 1620�D. 1800
MATHCOUNTS Chapter Lab — Lesson 7
Sprint Practice 7-8
Choose the best answer. Answers are not shown in this section.
7. A ratio is 2:3. If the total is 25, what is the smaller/larger first part?��A. 10�B. 15�C. 7�D. 12
8. What is 20% of 150?��A. 30�B. 750�C. 25�D. 35
MATHCOUNTS Chapter Lab — Lesson 7
Sprint Practice 9-10
Choose the best answer. Answers are not shown in this section.
9. The average of 16, 16, 22 and x is 20. What is x?��A. 30�B. 22�C. 26�D. 20
10. What is the Manhattan distance between (3,4) and (9,8)?��A. 10�B. 12�C. 4�D. 6
MATHCOUNTS Chapter Lab — Lesson 7
Sprint Practice 11-12
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11. A circle has radius 4. Its area is kπ. What is k?��A. 24�B. 4�C. 8�D. 16
12. A rectangle has length 12 and width 10. What is its area?��A. 120�B. 44�C. 132�D. 22
MATHCOUNTS Chapter Lab — Lesson 7
Sprint Practice 13-14
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13. A triangle has base 12 and height 8. What is its area?��A. 53�B. 96�C. 43�D. 48
14. A rectangular prism has dimensions 5, 7 and 6. What is its volume?��A. 18�B. 210�C. 216�D. 214
MATHCOUNTS Chapter Lab — Lesson 7
Sprint Practice 15-16
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15. A right triangle has legs 7 and 24. What is the hypotenuse?��A. 25�B. 17�C. 26�D. 31
16. What is the sum of the interior angles of a 8-sided polygon?��A. 1440�B. 900�C. 1080�D. 1260
MATHCOUNTS Chapter Lab — Lesson 7
Sprint Practice 17-18
Choose the best answer. Answers are not shown in this section.
17. A ratio is 2:5. If the total is 35, what is the smaller/larger first part?��A. 10�B. 12�C. 5�D. 25
18. What is 20% of 160?��A. 32�B. 27�C. 37�D. 800
MATHCOUNTS Chapter Lab — Lesson 7
Sprint Practice 19-20
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19. The average of 12, 16, 20 and x is 20. What is x?��A. 36�B. 28�C. 32�D. 20
20. What is the Manhattan distance between (3,2) and (9,7)?��A. 5�B. 11�C. 13�D. 6
MATHCOUNTS Chapter Lab — Lesson 7
Sprint Practice 21-22
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21. A circle has radius 7. Its area is kπ. What is k?��A. 7�B. 49�C. 63�D. 14
22. A rectangle has length 13 and width 9. What is its area?��A. 130�B. 44�C. 117�D. 22
MATHCOUNTS Chapter Lab — Lesson 7
Sprint Practice 23-24
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23. A triangle has base 16 and height 11. What is its area?��A. 88�B. 93�C. 83�D. 176
24. A rectangular prism has dimensions 5, 5 and 7. What is its volume?��A. 17�B. 190�C. 182�D. 175
MATHCOUNTS Chapter Lab — Lesson 7
Sprint Practice 25-26
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25. A right triangle has legs 7 and 24. What is the hypotenuse?��A. 31�B. 25�C. 26�D. 17
26. What is the sum of the interior angles of a 6-sided polygon?��A. 720�B. 540�C. 1080�D. 900
MATHCOUNTS Chapter Lab — Lesson 7
Sprint Practice 27-28
Choose the best answer. Answers are not shown in this section.
27. A ratio is 2:3. If the total is 25, what is the smaller/larger first part?��A. 10�B. 15�C. 12�D. 7
28. What is 20% of 80?��A. 11�B. 400�C. 21�D. 16
MATHCOUNTS Chapter Lab — Lesson 7
Sprint Practice 29-30
Choose the best answer. Answers are not shown in this section.
29. The average of 15, 16, 24 and x is 20. What is x?��A. 21�B. 20�C. 25�D. 29
30. What is the Manhattan distance between (3,4) and (9,10)?��A. 12�B. 14�C. 8�D. 6
MATHCOUNTS Chapter Lab — Lesson 7
Sprint Practice 31-32
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31. A circle has radius 3. Its area is kπ. What is k?��A. 15�B. 3�C. 6�D. 9
32. A rectangle has length 14 and width 8. What is its area?��A. 112�B. 22�C. 44�D. 126
MATHCOUNTS Chapter Lab — Lesson 7
Sprint Practice 33-34
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33. A triangle has base 20 and height 7. What is its area?��A. 65�B. 70�C. 140�D. 75
34. A rectangular prism has dimensions 5, 7 and 5. What is its volume?��A. 175�B. 180�C. 17�D. 190
MATHCOUNTS Chapter Lab — Lesson 7
Sprint Practice 35-36
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35. A right triangle has legs 7 and 24. What is the hypotenuse?��A. 26�B. 31�C. 25�D. 17
36. What is the sum of the interior angles of a 10-sided polygon?��A. 1440�B. 1260�C. 1800�D. 1620
MATHCOUNTS Chapter Lab — Lesson 7
Sprint Practice 37-38
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37. A ratio is 2:5. If the total is 35, what is the smaller/larger first part?��A. 5�B. 10�C. 12�D. 25
38. What is 20% of 90?��A. 450�B. 23�C. 18�D. 13
MATHCOUNTS Chapter Lab — Lesson 7
Sprint Practice 39-40
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39. The average of 11, 16, 22 and x is 20. What is x?��A. 35�B. 27�C. 20�D. 31
40. What is the Manhattan distance between (3,2) and (9,6)?��A. 6�B. 12�C. 4�D. 10
MATHCOUNTS Chapter Lab — Lesson 7
Sprint Practice 41-42
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41. A circle has radius 6. Its area is kπ. What is k?��A. 36�B. 48�C. 12�D. 6
42. A rectangle has length 15 and width 7. What is its area?��A. 22�B. 120�C. 105�D. 44
MATHCOUNTS Chapter Lab — Lesson 7
Sprint Practice 43-44
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43. A triangle has base 8 and height 10. What is its area?��A. 80�B. 40�C. 35�D. 45
44. A rectangular prism has dimensions 5, 5 and 6. What is its volume?��A. 150�B. 16�C. 156�D. 170
MATHCOUNTS Chapter Lab — Lesson 7
Sprint Practice 45-46
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45. A right triangle has legs 7 and 24. What is the hypotenuse?��A. 26�B. 25�C. 31�D. 17
46. What is the sum of the interior angles of a 8-sided polygon?��A. 900�B. 1260�C. 1440�D. 1080
MATHCOUNTS Chapter Lab — Lesson 7
Sprint Practice 47-48
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47. A ratio is 2:3. If the total is 25, what is the smaller/larger first part?��A. 15�B. 12�C. 7�D. 10
48. What is 20% of 100?��A. 500�B. 25�C. 15�D. 20
MATHCOUNTS Chapter Lab — Lesson 7
Sprint Practice 49-50
Choose the best answer. Answers are not shown in this section.
49. The average of 14, 16, 20 and x is 20. What is x?��A. 26�B. 30�C. 34�D. 20
50. What is the Manhattan distance between (3,4) and (9,9)?��A. 6�B. 13�C. 5�D. 11
MATHCOUNTS Chapter Lab — Lesson 7
Answer Key
Multiple-choice questions 1-50
1. C�2. B�3. A�4. C�5. D�6. B�7. A�8. A�9. C�10. A
11. D�12. A�13. D�14. B�15. A�16. C�17. A�18. A�19. C�20. B
21. B�22. C�23. A�24. D�25. B�26. A�27. A�28. D�29. C�30. A
31. D�32. A�33. B�34. A�35. C�36. A�37. B�38. C�39. D�40. D
41. A�42. C�43. B�44. A�45. B�46. D�47. D�48. D�49. B�50. D
MATHCOUNTS Chapter Lab — Lesson 7
Solutions 1-2
Read after completing the practice section.
1. Answer: C. 64�Circle area = πr^2. With radius 8, area = 64π, so k = 64.
2. Answer: B. 121�Area of a rectangle is length × width = 11 × 11 = 121.
MATHCOUNTS Chapter Lab — Lesson 7
Solutions 3-4
Read after completing the practice section.
3. Answer: A. 20�Triangle area = (1/2)bh = (1/2)(8)(5) = 20.
4. Answer: C. 125�Volume = length × width × height = 5×5×5 = 125.
MATHCOUNTS Chapter Lab — Lesson 7
Solutions 5-6
Read after completing the practice section.
5. Answer: D. 25�By the Pythagorean theorem, c^2 = 7^2+24^2 = 625, so c = 25.
6. 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.
MATHCOUNTS Chapter Lab — Lesson 7
Solutions 7-8
Read after completing the practice section.
7. Answer: A. 10�The total ratio parts are 5. One part is 25/5 = 5. The first part is 2 parts, or 10.
8. Answer: A. 30�20% = 20/100. So 20% of 150 is 20/100 × 150 = 30.
MATHCOUNTS Chapter Lab — Lesson 7
Solutions 9-10
Read after completing the practice section.
9. Answer: C. 26�Total for four numbers is 4×20 = 80. The known sum is 54, so x = 80-54 = 26.
10. Answer: A. 10�Manhattan distance is horizontal change plus vertical change: |9-3| + |8-4| = 10.
MATHCOUNTS Chapter Lab — Lesson 7
Solutions 11-12
Read after completing the practice section.
11. Answer: D. 16�Circle area = πr^2. With radius 4, area = 16π, so k = 16.
12. Answer: A. 120�Area of a rectangle is length × width = 12 × 10 = 120.
MATHCOUNTS Chapter Lab — Lesson 7
Solutions 13-14
Read after completing the practice section.
13. Answer: D. 48�Triangle area = (1/2)bh = (1/2)(12)(8) = 48.
14. Answer: B. 210�Volume = length × width × height = 5×7×6 = 210.
MATHCOUNTS Chapter Lab — Lesson 7
Solutions 15-16
Read after completing the practice section.
15. Answer: A. 25�By the Pythagorean theorem, c^2 = 7^2+24^2 = 625, so c = 25.
16. Answer: C. 1080�The sum of interior angles of an n-gon is (n-2)180. For n=8, the sum is (8-2)180 = 1080.
MATHCOUNTS Chapter Lab — Lesson 7
Solutions 17-18
Read after completing the practice section.
17. Answer: A. 10�The total ratio parts are 7. One part is 35/7 = 5. The first part is 2 parts, or 10.
18. Answer: A. 32�20% = 20/100. So 20% of 160 is 20/100 × 160 = 32.
MATHCOUNTS Chapter Lab — Lesson 7
Solutions 19-20
Read after completing the practice section.
19. Answer: C. 32�Total for four numbers is 4×20 = 80. The known sum is 48, so x = 80-48 = 32.
20. Answer: B. 11�Manhattan distance is horizontal change plus vertical change: |9-3| + |7-2| = 11.
MATHCOUNTS Chapter Lab — Lesson 7
Solutions 21-22
Read after completing the practice section.
21. Answer: B. 49�Circle area = πr^2. With radius 7, area = 49π, so k = 49.
22. Answer: C. 117�Area of a rectangle is length × width = 13 × 9 = 117.
MATHCOUNTS Chapter Lab — Lesson 7
Solutions 23-24
Read after completing the practice section.
23. Answer: A. 88�Triangle area = (1/2)bh = (1/2)(16)(11) = 88.
24. Answer: D. 175�Volume = length × width × height = 5×5×7 = 175.
MATHCOUNTS Chapter Lab — Lesson 7
Solutions 25-26
Read after completing the practice section.
25. Answer: B. 25�By the Pythagorean theorem, c^2 = 7^2+24^2 = 625, so c = 25.
26. Answer: A. 720�The sum of interior angles of an n-gon is (n-2)180. For n=6, the sum is (6-2)180 = 720.
MATHCOUNTS Chapter Lab — Lesson 7
Solutions 27-28
Read after completing the practice section.
27. Answer: A. 10�The total ratio parts are 5. One part is 25/5 = 5. The first part is 2 parts, or 10.
28. Answer: D. 16�20% = 20/100. So 20% of 80 is 20/100 × 80 = 16.
MATHCOUNTS Chapter Lab — Lesson 7
Solutions 29-30
Read after completing the practice section.
29. Answer: C. 25�Total for four numbers is 4×20 = 80. The known sum is 55, so x = 80-55 = 25.
30. Answer: A. 12�Manhattan distance is horizontal change plus vertical change: |9-3| + |10-4| = 12.
MATHCOUNTS Chapter Lab — Lesson 7
Solutions 31-32
Read after completing the practice section.
31. Answer: D. 9�Circle area = πr^2. With radius 3, area = 9π, so k = 9.
32. Answer: A. 112�Area of a rectangle is length × width = 14 × 8 = 112.
MATHCOUNTS Chapter Lab — Lesson 7
Solutions 33-34
Read after completing the practice section.
33. Answer: B. 70�Triangle area = (1/2)bh = (1/2)(20)(7) = 70.
34. Answer: A. 175�Volume = length × width × height = 5×7×5 = 175.
MATHCOUNTS Chapter Lab — Lesson 7
Solutions 35-36
Read after completing the practice section.
35. Answer: C. 25�By the Pythagorean theorem, c^2 = 7^2+24^2 = 625, so c = 25.
36. Answer: A. 1440�The sum of interior angles of an n-gon is (n-2)180. For n=10, the sum is (10-2)180 = 1440.
MATHCOUNTS Chapter Lab — Lesson 7
Solutions 37-38
Read after completing the practice section.
37. Answer: B. 10�The total ratio parts are 7. One part is 35/7 = 5. The first part is 2 parts, or 10.
38. Answer: C. 18�20% = 20/100. So 20% of 90 is 20/100 × 90 = 18.
MATHCOUNTS Chapter Lab — Lesson 7
Solutions 39-40
Read after completing the practice section.
39. Answer: D. 31�Total for four numbers is 4×20 = 80. The known sum is 49, so x = 80-49 = 31.
40. Answer: D. 10�Manhattan distance is horizontal change plus vertical change: |9-3| + |6-2| = 10.
MATHCOUNTS Chapter Lab — Lesson 7
Solutions 41-42
Read after completing the practice section.
41. Answer: A. 36�Circle area = πr^2. With radius 6, area = 36π, so k = 36.
42. Answer: C. 105�Area of a rectangle is length × width = 15 × 7 = 105.
MATHCOUNTS Chapter Lab — Lesson 7
Solutions 43-44
Read after completing the practice section.
43. Answer: B. 40�Triangle area = (1/2)bh = (1/2)(8)(10) = 40.
44. Answer: A. 150�Volume = length × width × height = 5×5×6 = 150.
MATHCOUNTS Chapter Lab — Lesson 7
Solutions 45-46
Read after completing the practice section.
45. Answer: B. 25�By the Pythagorean theorem, c^2 = 7^2+24^2 = 625, so c = 25.
46. Answer: D. 1080�The sum of interior angles of an n-gon is (n-2)180. For n=8, the sum is (8-2)180 = 1080.
MATHCOUNTS Chapter Lab — Lesson 7
Solutions 47-48
Read after completing the practice section.
47. Answer: D. 10�The total ratio parts are 5. One part is 25/5 = 5. The first part is 2 parts, or 10.
48. Answer: D. 20�20% = 20/100. So 20% of 100 is 20/100 × 100 = 20.
MATHCOUNTS Chapter Lab — Lesson 7
Solutions 49-50
Read after completing the practice section.
49. Answer: B. 30�Total for four numbers is 4×20 = 80. The known sum is 50, so x = 80-50 = 30.
50. Answer: D. 11�Manhattan distance is horizontal change plus vertical change: |9-3| + |9-4| = 11.
MATHCOUNTS Chapter Lab — Lesson 7