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Grade 12 · Maths · curriculum collection

Grade 12 Maths, Common Core conceptual categories

8 units for Grade 12 maths, organised by Common Core conceptual category: Algebra, Functions, Geometry, Statistics and Probability. Each unit has a short lesson, a practice worksheet and a review worksheet, both with answer keys.

8 units16 worksheets192 questions with answers
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Answer-checked and approved on 29 October 2026

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Aligned by StudyBrix at domain level. Not an official publication of, or endorsed by, the standards' authors. Framework: Common Core State Standards for Mathematics.

Units in teaching order

01

HS.A Algebra

Quadratic expressions and equations

The discriminant and completing the square.

Short lesson: Complete the square

Study guide for Grades 9 to 12: 2 worked examples and 2 practice questions. Halve the coefficient of x, then square it.

x² + 2hx + h² = (x + h)²

Halve the coefficient of x, then square it.

Worked example

Complete x² + 68x + □ = (x + 34)².

(68 ÷ 2)² = 1156

Answer: 1156

Read the full study guide

Worksheets

A practice worksheet and a review worksheet, 12 and 12 questions.

Worksheets (PDF)Answer keys (PDF)
Practice worksheet: 12 questions

  1. Find the discriminant of 8x² + 11x + 15 = 0.

  2. Complete x² + 38x + □ = (x + 19)².

  3. Find the discriminant of 2x² + 8x + 17 = 0.

  4. Complete x² + 60x + □ = (x + 30)².

  5. Find the discriminant of 2x² + 15x + 1 = 0.

  6. Complete x² + 62x + □ = (x + 31)².

  7. Find the discriminant of 2x² + 16x + 5 = 0.

  8. Complete x² + 8x + □ = (x + 4)².

  9. Find the discriminant of 8x² + 18x + 18 = 0.

  10. Complete x² + 56x + □ = (x + 28)².

  11. Find the discriminant of 8x² + 4x + 5 = 0.

  12. Complete x² + 80x + □ = (x + 40)².

Answer key
  1. -359 11² − 4 × 8 × 15 = -359
  2. 361 (38 ÷ 2)² = 361
  3. -72 8² − 4 × 2 × 17 = -72
  4. 900 (60 ÷ 2)² = 900
  5. 217 15² − 4 × 2 × 1 = 217
  6. 961 (62 ÷ 2)² = 961
  7. 216 16² − 4 × 2 × 5 = 216
  8. 16 (8 ÷ 2)² = 16
  9. -252 18² − 4 × 8 × 18 = -252
  10. 784 (56 ÷ 2)² = 784
  11. -144 4² − 4 × 8 × 5 = -144
  12. 1600 (80 ÷ 2)² = 1600
Review worksheet: 12 questions

  1. Find the discriminant of 8x² + 18x + 2 = 0.

  2. Complete x² + 68x + □ = (x + 34)².

  3. Find the discriminant of 1x² + 16x + 8 = 0.

  4. Complete x² + 48x + □ = (x + 24)².

  5. Find the discriminant of 6x² + 16x + 17 = 0.

  6. Complete x² + 26x + □ = (x + 13)².

  7. Find the discriminant of 7x² + 1x + 19 = 0.

  8. Complete x² + 66x + □ = (x + 33)².

  9. Find the discriminant of 4x² + 1x + 7 = 0.

  10. Complete x² + 30x + □ = (x + 15)².

  11. Find the discriminant of 3x² + 20x + 2 = 0.

  12. Complete x² + 36x + □ = (x + 18)².

Answer key
  1. 260 18² − 4 × 8 × 2 = 260
  2. 1156 (68 ÷ 2)² = 1156
  3. 224 16² − 4 × 1 × 8 = 224
  4. 576 (48 ÷ 2)² = 576
  5. -152 16² − 4 × 6 × 17 = -152
  6. 169 (26 ÷ 2)² = 169
  7. -531 1² − 4 × 7 × 19 = -531
  8. 1089 (66 ÷ 2)² = 1089
  9. -111 1² − 4 × 4 × 7 = -111
  10. 225 (30 ÷ 2)² = 225
  11. 376 20² − 4 × 3 × 2 = 376
  12. 324 (36 ÷ 2)² = 324

Related ready-made worksheets: Grade 12 Quadratic equations worksheet

02

HS.A Algebra

Geometric series

Sums of geometric series.

Short lesson: Finite geometric sums

Study guide for Grades 10 to 12: 2 worked examples and 2 practice questions. Use n terms, starting with the first term a.

Sₙ = a(rⁿ − 1)/(r − 1), r ≠ 1

Use n terms, starting with the first term a.

Worked example

Find the sum of the first 4 terms of a geometric sequence with first term 4 and ratio 4.

4 × (4^4 − 1) ÷ (4 − 1) = 340

Answer: 340

Read the full study guide

Worksheets

A practice worksheet and a review worksheet, 12 and 12 questions.

Worksheets (PDF)Answer keys (PDF)
Practice worksheet: 12 questions

  1. Find the sum of the first 5 terms of a geometric sequence with first term 5 and ratio 4.

  2. Find the sum of the first 3 terms of a geometric sequence with first term 15 and ratio 3.

  3. Find the sum of the first 6 terms of a geometric sequence with first term 2 and ratio 4.

  4. Find the sum of the first 6 terms of a geometric sequence with first term 15 and ratio 5.

  5. Find the sum of the first 3 terms of a geometric sequence with first term 2 and ratio 4.

  6. Find the sum of the first 4 terms of a geometric sequence with first term 7 and ratio 4.

  7. Find the sum of the first 6 terms of a geometric sequence with first term 4 and ratio 4.

  8. Find the sum of the first 7 terms of a geometric sequence with first term 6 and ratio 2.

  9. Find the sum of the first 7 terms of a geometric sequence with first term 10 and ratio 2.

  10. Find the sum of the first 7 terms of a geometric sequence with first term 7 and ratio 5.

  11. Find the sum of the first 3 terms of a geometric sequence with first term 10 and ratio 5.

  12. Find the sum of the first 6 terms of a geometric sequence with first term 6 and ratio 5.

Answer key
  1. 1705 5 × (4^5 − 1) ÷ (4 − 1) = 1705
  2. 195 15 × (3^3 − 1) ÷ (3 − 1) = 195
  3. 2730 2 × (4^6 − 1) ÷ (4 − 1) = 2730
  4. 58590 15 × (5^6 − 1) ÷ (5 − 1) = 58590
  5. 42 2 × (4^3 − 1) ÷ (4 − 1) = 42
  6. 595 7 × (4^4 − 1) ÷ (4 − 1) = 595
  7. 5460 4 × (4^6 − 1) ÷ (4 − 1) = 5460
  8. 762 6 × (2^7 − 1) ÷ (2 − 1) = 762
  9. 1270 10 × (2^7 − 1) ÷ (2 − 1) = 1270
  10. 136717 7 × (5^7 − 1) ÷ (5 − 1) = 136717
  11. 310 10 × (5^3 − 1) ÷ (5 − 1) = 310
  12. 23436 6 × (5^6 − 1) ÷ (5 − 1) = 23436
Review worksheet: 12 questions

  1. Find the sum of the first 5 terms of a geometric sequence with first term 5 and ratio 2.

  2. Find the sum of the first 4 terms of a geometric sequence with first term 6 and ratio 3.

  3. Find the sum of the first 6 terms of a geometric sequence with first term 1 and ratio 2.

  4. Find the sum of the first 3 terms of a geometric sequence with first term 9 and ratio 3.

  5. Find the sum of the first 3 terms of a geometric sequence with first term 12 and ratio 5.

  6. Find the sum of the first 6 terms of a geometric sequence with first term 13 and ratio 5.

  7. Find the sum of the first 4 terms of a geometric sequence with first term 6 and ratio 4.

  8. Find the sum of the first 5 terms of a geometric sequence with first term 11 and ratio 5.

  9. Find the sum of the first 6 terms of a geometric sequence with first term 12 and ratio 3.

  10. Find the sum of the first 7 terms of a geometric sequence with first term 11 and ratio 5.

  11. Find the sum of the first 6 terms of a geometric sequence with first term 6 and ratio 2.

  12. Find the sum of the first 5 terms of a geometric sequence with first term 7 and ratio 4.

Answer key
  1. 155 5 × (2^5 − 1) ÷ (2 − 1) = 155
  2. 240 6 × (3^4 − 1) ÷ (3 − 1) = 240
  3. 63 1 × (2^6 − 1) ÷ (2 − 1) = 63
  4. 117 9 × (3^3 − 1) ÷ (3 − 1) = 117
  5. 372 12 × (5^3 − 1) ÷ (5 − 1) = 372
  6. 50778 13 × (5^6 − 1) ÷ (5 − 1) = 50778
  7. 510 6 × (4^4 − 1) ÷ (4 − 1) = 510
  8. 8591 11 × (5^5 − 1) ÷ (5 − 1) = 8591
  9. 4368 12 × (3^6 − 1) ÷ (3 − 1) = 4368
  10. 214841 11 × (5^7 − 1) ÷ (5 − 1) = 214841
  11. 378 6 × (2^6 − 1) ÷ (2 − 1) = 378
  12. 2387 7 × (4^5 − 1) ÷ (4 − 1) = 2387

Related ready-made worksheets: Grade 12 Sequences & series worksheet

03

HS.F Functions

Inverse and composite functions

Composing functions.

Short lesson: Compose two functions

Study guide for Grades 9 to 12: 2 worked examples and 2 practice questions. Apply the inside function first.

(f ∘ g)(x) = f(g(x))

Apply the inside function first.

Worked example

f(t) = 4t + (-3); g(t) = 5t + (-5). Find f(g(7)).

g(7) = 30; f(30) = 117

Answer: 117

Read the full study guide

Worksheets

A practice worksheet and a review worksheet, 12 and 12 questions.

Worksheets (PDF)Answer keys (PDF)
Practice worksheet: 12 questions

  1. f(t) = 2t + (1); g(t) = 6t + (0). Find f(g(-3)).

  2. f(t) = 3t + (-7); g(t) = 2t + (6). Find f(g(-3)).

  3. f(t) = 3t + (-10); g(t) = 3t + (-4). Find f(g(3)).

  4. f(t) = 6t + (-3); g(t) = 8t + (6). Find f(g(2)).

  5. f(t) = 7t + (-9); g(t) = 8t + (-4). Find f(g(-3)).

  6. f(t) = 3t + (1); g(t) = 2t + (-8). Find f(g(0)).

  7. f(t) = 9t + (5); g(t) = 8t + (-6). Find f(g(4)).

  8. f(t) = 8t + (-8); g(t) = 2t + (-4). Find f(g(-3)).

  9. f(t) = 3t + (-10); g(t) = 2t + (6). Find f(g(3)).

  10. f(t) = 7t + (-1); g(t) = 6t + (-2). Find f(g(-3)).

  11. f(t) = 5t + (3); g(t) = 1t + (-6). Find f(g(-1)).

  12. f(t) = 7t + (-4); g(t) = 6t + (6). Find f(g(-2)).

Answer key
  1. -35 g(-3) = -18; f(-18) = -35
  2. -7 g(-3) = 0; f(0) = -7
  3. 5 g(3) = 5; f(5) = 5
  4. 129 g(2) = 22; f(22) = 129
  5. -205 g(-3) = -28; f(-28) = -205
  6. -23 g(0) = -8; f(-8) = -23
  7. 239 g(4) = 26; f(26) = 239
  8. -88 g(-3) = -10; f(-10) = -88
  9. 26 g(3) = 12; f(12) = 26
  10. -141 g(-3) = -20; f(-20) = -141
  11. -32 g(-1) = -7; f(-7) = -32
  12. -46 g(-2) = -6; f(-6) = -46
Review worksheet: 12 questions

  1. f(t) = 7t + (10); g(t) = 3t + (8). Find f(g(8)).

  2. f(t) = 5t + (-1); g(t) = 6t + (5). Find f(g(3)).

  3. f(t) = 2t + (-4); g(t) = 2t + (3). Find f(g(-1)).

  4. f(t) = 7t + (-5); g(t) = 2t + (-8). Find f(g(2)).

  5. f(t) = 8t + (1); g(t) = 4t + (-8). Find f(g(-1)).

  6. f(t) = 9t + (3); g(t) = 8t + (1). Find f(g(6)).

  7. f(t) = 5t + (2); g(t) = 3t + (7). Find f(g(0)).

  8. f(t) = 2t + (0); g(t) = 5t + (-2). Find f(g(5)).

  9. f(t) = 9t + (-1); g(t) = 8t + (4). Find f(g(4)).

  10. f(t) = 5t + (-9); g(t) = 4t + (-4). Find f(g(0)).

  11. f(t) = 4t + (0); g(t) = 2t + (8). Find f(g(2)).

  12. f(t) = 4t + (1); g(t) = 3t + (4). Find f(g(5)).

Answer key
  1. 234 g(8) = 32; f(32) = 234
  2. 114 g(3) = 23; f(23) = 114
  3. -2 g(-1) = 1; f(1) = -2
  4. -33 g(2) = -4; f(-4) = -33
  5. -95 g(-1) = -12; f(-12) = -95
  6. 444 g(6) = 49; f(49) = 444
  7. 37 g(0) = 7; f(7) = 37
  8. 46 g(5) = 23; f(23) = 46
  9. 323 g(4) = 36; f(36) = 323
  10. -29 g(0) = -4; f(-4) = -29
  11. 48 g(2) = 12; f(12) = 48
  12. 77 g(5) = 19; f(19) = 77

Related ready-made worksheets: Grade 12 Algebra worksheet

04

HS.F Functions

Quadratic functions

The axis of symmetry of a parabola.

Short lesson: Axis of symmetry

Study guide for Grades 10 to 12: 2 worked examples and 2 practice questions. The vertex lies on this vertical axis for y = ax² + bx + c.

x = −b/(2a), a ≠ 0

The vertex lies on this vertical axis for y = ax² + bx + c.

Worked example

The graph y = 1x² + 33x + 7 has axis x = k. Find k.

k = −33/(2 × 1) = -33/2

Answer: -33/2

Read the full study guide

Worksheets

A practice worksheet and a review worksheet, 12 and 12 questions.

Worksheets (PDF)Answer keys (PDF)
Practice worksheet: 12 questions

  1. The graph y = 10x² + 1x + 18 has axis x = k. Find k.

  2. The graph y = 3x² + 38x + 1 has axis x = k. Find k.

  3. The graph y = 6x² + 11x + 8 has axis x = k. Find k.

  4. The graph y = 3x² + 37x + 11 has axis x = k. Find k.

  5. The graph y = 9x² + 8x + 12 has axis x = k. Find k.

  6. The graph y = 5x² + 17x + 18 has axis x = k. Find k.

  7. The graph y = 6x² + 12x + 20 has axis x = k. Find k.

  8. The graph y = 2x² + 3x + 15 has axis x = k. Find k.

  9. The graph y = 9x² + 20x + 15 has axis x = k. Find k.

  10. The graph y = 1x² + 12x + 13 has axis x = k. Find k.

  11. The graph y = 8x² + 32x + 16 has axis x = k. Find k.

  12. The graph y = 4x² + 18x + 20 has axis x = k. Find k.

Answer key
  1. -1/20 k = −1/(2 × 10) = -1/20
  2. -19/3 k = −38/(2 × 3) = -19/3
  3. -11/12 k = −11/(2 × 6) = -11/12
  4. -37/6 k = −37/(2 × 3) = -37/6
  5. -4/9 k = −8/(2 × 9) = -4/9
  6. -17/10 k = −17/(2 × 5) = -17/10
  7. -1 k = −12/(2 × 6) = -1
  8. -3/4 k = −3/(2 × 2) = -3/4
  9. -10/9 k = −20/(2 × 9) = -10/9
  10. -6 k = −12/(2 × 1) = -6
  11. -2 k = −32/(2 × 8) = -2
  12. -9/4 k = −18/(2 × 4) = -9/4
Review worksheet: 12 questions

  1. The graph y = 8x² + 37x + 3 has axis x = k. Find k.

  2. The graph y = 3x² + 37x + 15 has axis x = k. Find k.

  3. The graph y = 7x² + 18x + 12 has axis x = k. Find k.

  4. The graph y = 2x² + 31x + 11 has axis x = k. Find k.

  5. The graph y = 5x² + 10x + 12 has axis x = k. Find k.

  6. The graph y = 4x² + 15x + 19 has axis x = k. Find k.

  7. The graph y = 1x² + 37x + 17 has axis x = k. Find k.

  8. The graph y = 2x² + 1x + 1 has axis x = k. Find k.

  9. The graph y = 2x² + 15x + 17 has axis x = k. Find k.

  10. The graph y = 8x² + 40x + 3 has axis x = k. Find k.

  11. The graph y = 2x² + 15x + 1 has axis x = k. Find k.

  12. The graph y = 7x² + 3x + 4 has axis x = k. Find k.

Answer key
  1. -37/16 k = −37/(2 × 8) = -37/16
  2. -37/6 k = −37/(2 × 3) = -37/6
  3. -9/7 k = −18/(2 × 7) = -9/7
  4. -31/4 k = −31/(2 × 2) = -31/4
  5. -1 k = −10/(2 × 5) = -1
  6. -15/8 k = −15/(2 × 4) = -15/8
  7. -37/2 k = −37/(2 × 1) = -37/2
  8. -1/4 k = −1/(2 × 2) = -1/4
  9. -15/4 k = −15/(2 × 2) = -15/4
  10. -5/2 k = −40/(2 × 8) = -5/2
  11. -15/4 k = −15/(2 × 2) = -15/4
  12. -3/14 k = −3/(2 × 7) = -3/14

Related ready-made worksheets: Grade 12 Quadratic equations worksheet

05

HS.F Functions

Sequences, logarithms and radians

Arithmetic sequences and geometric sequences.

Short lesson: Arithmetic sequences

Study guide for Grades 9 to 12: 2 worked examples and 2 practice questions. The first term is n = 1. Use one fewer step than the term number.

aₙ = a₁ + (n − 1)d

The first term is n = 1. Use one fewer step than the term number.

Worked example

An arithmetic sequence has first term 17 and common difference 2. Find term 24.

17 + (24 − 1) × 2 = 63

Answer: 63

Read the full study guide

Worksheets

A practice worksheet and a review worksheet, 12 and 12 questions.

Worksheets (PDF)Answer keys (PDF)
Practice worksheet: 12 questions

  1. An arithmetic sequence has first term 3 and common difference 11. Find term 19.

  2. A geometric sequence starts at 11, with common ratio 2. Find term 4.

  3. An arithmetic sequence has first term 31 and common difference 14. Find term 16.

  4. A geometric sequence starts at 10, with common ratio 3. Find term 5.

  5. An arithmetic sequence has first term 40 and common difference 12. Find term 5.

  6. A geometric sequence starts at 10, with common ratio 2. Find term 3.

  7. An arithmetic sequence has first term 2 and common difference 3. Find term 17.

  8. A geometric sequence starts at 1, with common ratio 2. Find term 6.

  9. An arithmetic sequence has first term 36 and common difference 14. Find term 14.

  10. A geometric sequence starts at 5, with common ratio 4. Find term 6.

  11. An arithmetic sequence has first term 39 and common difference 5. Find term 27.

  12. A geometric sequence starts at 9, with common ratio 5. Find term 5.

Answer key
  1. 201 3 + (19 − 1) × 11 = 201
  2. 88 11 × 2^(4 − 1) = 88
  3. 241 31 + (16 − 1) × 14 = 241
  4. 810 10 × 3^(5 − 1) = 810
  5. 88 40 + (5 − 1) × 12 = 88
  6. 40 10 × 2^(3 − 1) = 40
  7. 50 2 + (17 − 1) × 3 = 50
  8. 32 1 × 2^(6 − 1) = 32
  9. 218 36 + (14 − 1) × 14 = 218
  10. 5120 5 × 4^(6 − 1) = 5120
  11. 169 39 + (27 − 1) × 5 = 169
  12. 5625 9 × 5^(5 − 1) = 5625
Review worksheet: 12 questions

  1. An arithmetic sequence has first term 36 and common difference 9. Find term 17.

  2. A geometric sequence starts at 11, with common ratio 2. Find term 6.

  3. An arithmetic sequence has first term 15 and common difference 9. Find term 10.

  4. A geometric sequence starts at 3, with common ratio 3. Find term 4.

  5. An arithmetic sequence has first term 40 and common difference 4. Find term 25.

  6. A geometric sequence starts at 4, with common ratio 2. Find term 4.

  7. An arithmetic sequence has first term 20 and common difference 12. Find term 16.

  8. A geometric sequence starts at 2, with common ratio 2. Find term 5.

  9. An arithmetic sequence has first term 24 and common difference 2. Find term 18.

  10. A geometric sequence starts at 5, with common ratio 3. Find term 6.

  11. An arithmetic sequence has first term 34 and common difference 7. Find term 7.

  12. A geometric sequence starts at 5, with common ratio 2. Find term 6.

Answer key
  1. 180 36 + (17 − 1) × 9 = 180
  2. 352 11 × 2^(6 − 1) = 352
  3. 96 15 + (10 − 1) × 9 = 96
  4. 81 3 × 3^(4 − 1) = 81
  5. 136 40 + (25 − 1) × 4 = 136
  6. 32 4 × 2^(4 − 1) = 32
  7. 200 20 + (16 − 1) × 12 = 200
  8. 32 2 × 2^(5 − 1) = 32
  9. 58 24 + (18 − 1) × 2 = 58
  10. 1215 5 × 3^(6 − 1) = 1215
  11. 76 34 + (7 − 1) × 7 = 76
  12. 160 5 × 2^(6 − 1) = 160

Related ready-made worksheets: Grade 12 Sequences & series worksheet

06

HS.G Geometry

Geometry

Volume of cones.

Short lesson: Volume of a cone

Study guide for Grades 8 to 12: 2 worked examples and 2 practice questions. Use the perpendicular height and radius, not the diameter.

V = πr²h/3

Use the perpendicular height and radius, not the diameter.

Worked example

A cone has radius 2 cm and height 90 cm. Use π = 3.14. Find its volume in cm³.

3.14 × 2² × 90 ÷ 3 = 376.8

Answer: 376.8 cm³

Read the full study guide

Worksheets

A practice worksheet and a review worksheet, 12 and 12 questions.

Worksheets (PDF)Answer keys (PDF)
Practice worksheet: 12 questions

  1. A cone has radius 8 cm and height 63 cm. Use π = 3.14. Find its volume in cm³.

  2. A cone has radius 3 cm and height 30 cm. Use π = 3.14. Find its volume in cm³.

  3. A cone has radius 7 cm and height 90 cm. Use π = 3.14. Find its volume in cm³.

  4. A cone has radius 21 cm and height 30 cm. Use π = 3.14. Find its volume in cm³.

  5. A cone has radius 14 cm and height 66 cm. Use π = 3.14. Find its volume in cm³.

  6. A cone has radius 10 cm and height 51 cm. Use π = 3.14. Find its volume in cm³.

  7. A cone has radius 11 cm and height 63 cm. Use π = 3.14. Find its volume in cm³.

  8. A cone has radius 20 cm and height 63 cm. Use π = 3.14. Find its volume in cm³.

  9. A cone has radius 18 cm and height 33 cm. Use π = 3.14. Find its volume in cm³.

  10. A cone has radius 13 cm and height 57 cm. Use π = 3.14. Find its volume in cm³.

  11. A cone has radius 10 cm and height 6 cm. Use π = 3.14. Find its volume in cm³.

  12. A cone has radius 25 cm and height 75 cm. Use π = 3.14. Find its volume in cm³.

Answer key
  1. 4220.16 cm³ 3.14 × 8² × 63 ÷ 3 = 4220.16
  2. 282.6 cm³ 3.14 × 3² × 30 ÷ 3 = 282.6
  3. 4615.8 cm³ 3.14 × 7² × 90 ÷ 3 = 4615.8
  4. 13847.4 cm³ 3.14 × 21² × 30 ÷ 3 = 13847.4
  5. 13539.68 cm³ 3.14 × 14² × 66 ÷ 3 = 13539.68
  6. 5338 cm³ 3.14 × 10² × 51 ÷ 3 = 5338
  7. 7978.74 cm³ 3.14 × 11² × 63 ÷ 3 = 7978.74
  8. 26376 cm³ 3.14 × 20² × 63 ÷ 3 = 26376
  9. 11190.96 cm³ 3.14 × 18² × 33 ÷ 3 = 11190.96
  10. 10082.54 cm³ 3.14 × 13² × 57 ÷ 3 = 10082.54
  11. 628 cm³ 3.14 × 10² × 6 ÷ 3 = 628
  12. 49062.5 cm³ 3.14 × 25² × 75 ÷ 3 = 49062.5
Review worksheet: 12 questions

  1. A cone has radius 12 cm and height 78 cm. Use π = 3.14. Find its volume in cm³.

  2. A cone has radius 23 cm and height 90 cm. Use π = 3.14. Find its volume in cm³.

  3. A cone has radius 13 cm and height 69 cm. Use π = 3.14. Find its volume in cm³.

  4. A cone has radius 11 cm and height 21 cm. Use π = 3.14. Find its volume in cm³.

  5. A cone has radius 7 cm and height 60 cm. Use π = 3.14. Find its volume in cm³.

  6. A cone has radius 9 cm and height 6 cm. Use π = 3.14. Find its volume in cm³.

  7. A cone has radius 22 cm and height 21 cm. Use π = 3.14. Find its volume in cm³.

  8. A cone has radius 22 cm and height 78 cm. Use π = 3.14. Find its volume in cm³.

  9. A cone has radius 15 cm and height 21 cm. Use π = 3.14. Find its volume in cm³.

  10. A cone has radius 3 cm and height 63 cm. Use π = 3.14. Find its volume in cm³.

  11. A cone has radius 14 cm and height 81 cm. Use π = 3.14. Find its volume in cm³.

  12. A cone has radius 5 cm and height 90 cm. Use π = 3.14. Find its volume in cm³.

Answer key
  1. 11756.16 cm³ 3.14 × 12² × 78 ÷ 3 = 11756.16
  2. 49831.8 cm³ 3.14 × 23² × 90 ÷ 3 = 49831.8
  3. 12205.18 cm³ 3.14 × 13² × 69 ÷ 3 = 12205.18
  4. 2659.58 cm³ 3.14 × 11² × 21 ÷ 3 = 2659.58
  5. 3077.2 cm³ 3.14 × 7² × 60 ÷ 3 = 3077.2
  6. 508.68 cm³ 3.14 × 9² × 6 ÷ 3 = 508.68
  7. 10638.32 cm³ 3.14 × 22² × 21 ÷ 3 = 10638.32
  8. 39513.76 cm³ 3.14 × 22² × 78 ÷ 3 = 39513.76
  9. 4945.5 cm³ 3.14 × 15² × 21 ÷ 3 = 4945.5
  10. 593.46 cm³ 3.14 × 3² × 63 ÷ 3 = 593.46
  11. 16616.88 cm³ 3.14 × 14² × 81 ÷ 3 = 16616.88
  12. 2355 cm³ 3.14 × 5² × 90 ÷ 3 = 2355
07

HS.G Geometry

Right-triangle trigonometry

The tangent ratio and the sine ratio.

Short lesson: Sine in a right triangle

Study guide for Grades 9 to 12: 2 worked examples and 2 practice questions. Name the angle first; opposite is the side across from that angle.

sin θ = opposite / hypotenuse

Name the angle first; opposite is the side across from that angle.

Worked example

In a right triangle, the side opposite θ is 165 cm and the hypotenuse is 429 cm. Find sin θ as a fraction.

165/429 = 5/13

Answer: 5/13

Read the full study guide

Worksheets

A practice worksheet and a review worksheet, 12 and 12 questions.

Worksheets (PDF)Answer keys (PDF)
Practice worksheet: 12 questions

  1. In a right triangle, the legs opposite and adjacent to θ are 32 cm and 35 cm. Find tan θ as a fraction.

  2. In a right triangle, the side opposite θ is 5 cm and the hypotenuse is 13 cm. Find sin θ as a fraction.

  3. In a right triangle, the legs opposite and adjacent to θ are 2 cm and 8 cm. Find tan θ as a fraction.

  4. In a right triangle, the side opposite θ is 24 cm and the hypotenuse is 40 cm. Find sin θ as a fraction.

  5. In a right triangle, the legs opposite and adjacent to θ are 34 cm and 36 cm. Find tan θ as a fraction.

  6. In a right triangle, the side opposite θ is 272 cm and the hypotenuse is 578 cm. Find sin θ as a fraction.

  7. In a right triangle, the legs opposite and adjacent to θ are 35 cm and 5 cm. Find tan θ as a fraction.

  8. In a right triangle, the side opposite θ is 190 cm and the hypotenuse is 494 cm. Find sin θ as a fraction.

  9. In a right triangle, the legs opposite and adjacent to θ are 23 cm and 17 cm. Find tan θ as a fraction.

  10. In a right triangle, the side opposite θ is 144 cm and the hypotenuse is 306 cm. Find sin θ as a fraction.

  11. In a right triangle, the legs opposite and adjacent to θ are 37 cm and 24 cm. Find tan θ as a fraction.

  12. In a right triangle, the side opposite θ is 90 cm and the hypotenuse is 234 cm. Find sin θ as a fraction.

Answer key
  1. 32/35 32/35 = 32/35
  2. 5/13 5/13 = 5/13
  3. 1/4 2/8 = 1/4
  4. 3/5 24/40 = 3/5
  5. 17/18 34/36 = 17/18
  6. 8/17 272/578 = 8/17
  7. 7 35/5 = 7
  8. 5/13 190/494 = 5/13
  9. 23/17 23/17 = 23/17
  10. 8/17 144/306 = 8/17
  11. 37/24 37/24 = 37/24
  12. 5/13 90/234 = 5/13
Review worksheet: 12 questions

  1. In a right triangle, the legs opposite and adjacent to θ are 3 cm and 24 cm. Find tan θ as a fraction.

  2. In a right triangle, the side opposite θ is 80 cm and the hypotenuse is 208 cm. Find sin θ as a fraction.

  3. In a right triangle, the legs opposite and adjacent to θ are 7 cm and 32 cm. Find tan θ as a fraction.

  4. In a right triangle, the side opposite θ is 95 cm and the hypotenuse is 247 cm. Find sin θ as a fraction.

  5. In a right triangle, the legs opposite and adjacent to θ are 38 cm and 9 cm. Find tan θ as a fraction.

  6. In a right triangle, the side opposite θ is 140 cm and the hypotenuse is 364 cm. Find sin θ as a fraction.

  7. In a right triangle, the legs opposite and adjacent to θ are 36 cm and 38 cm. Find tan θ as a fraction.

  8. In a right triangle, the side opposite θ is 75 cm and the hypotenuse is 125 cm. Find sin θ as a fraction.

  9. In a right triangle, the legs opposite and adjacent to θ are 31 cm and 20 cm. Find tan θ as a fraction.

  10. In a right triangle, the side opposite θ is 3 cm and the hypotenuse is 5 cm. Find sin θ as a fraction.

  11. In a right triangle, the legs opposite and adjacent to θ are 11 cm and 10 cm. Find tan θ as a fraction.

  12. In a right triangle, the side opposite θ is 50 cm and the hypotenuse is 130 cm. Find sin θ as a fraction.

Answer key
  1. 1/8 3/24 = 1/8
  2. 5/13 80/208 = 5/13
  3. 7/32 7/32 = 7/32
  4. 5/13 95/247 = 5/13
  5. 38/9 38/9 = 38/9
  6. 5/13 140/364 = 5/13
  7. 18/19 36/38 = 18/19
  8. 3/5 75/125 = 3/5
  9. 31/20 31/20 = 31/20
  10. 3/5 3/5 = 3/5
  11. 11/10 11/10 = 11/10
  12. 5/13 50/130 = 5/13

Related ready-made worksheets: Grade 12 Trigonometry worksheet

08

HS.S Statistics and Probability

Statistics & probability

The mean of a frequency table, conditional probability from a table, variance, interquartile range, combinations and permutations.

Short lesson: Conditional probability from counts

Study guide for Grades 9 to 12: 2 worked examples and 2 practice questions. Restrict the denominator to the group named in the condition.

P(A | B) = count(A and B) / count(B)

Restrict the denominator to the group named in the condition.

Worked example

Club A has 6 bus users and 19 walkers. Club B has 16 bus users. Given that a student is in Club A, what is the probability they use the bus? Give a simplified fraction.

6/(6 + 19) = 6/25

Answer: 6/25

Read the full study guide

Worksheets

A practice worksheet and a review worksheet, 12 and 12 questions.

Worksheets (PDF)Answer keys (PDF)
Practice worksheet: 12 questions

  1. A score of 22 occurs 6 times; a score of 5 occurs 3 times. Find the mean as a fraction or decimal.

  2. Data: 7, 15, 23, 31, 39, 47, 55, 63. Using medians of the lower four and upper four values, find the interquartile range.

  3. How many ordered selections of 2 distinct items can be made from 12 items without repetition?

  4. Find the population variance of 41, 42, 43, 44, 45. Divide by 5.

  5. Club A has 9 bus users and 3 walkers. Club B has 3 bus users. Given that a student is in Club A, what is the probability they use the bus? Give a simplified fraction.

  6. How many different groups of 4 can be chosen from 5 distinct people, if order does not matter?

  7. A score of 19 occurs 10 times; a score of 8 occurs 4 times. Find the mean as a fraction or decimal.

  8. Data: 20, 32, 44, 56, 68, 80, 92, 104. Using medians of the lower four and upper four values, find the interquartile range.

  9. How many ordered selections of 5 distinct items can be made from 14 items without repetition?

  10. Find the population variance of 25, 28, 31, 34, 37. Divide by 5.

  11. Club A has 14 bus users and 4 walkers. Club B has 11 bus users. Given that a student is in Club A, what is the probability they use the bus? Give a simplified fraction.

  12. How many different groups of 2 can be chosen from 14 distinct people, if order does not matter?

Answer key
  1. 49/3 (22 × 6 + 5 × 3)/(6 + 3) = 49/3
  2. 32 Q₁ = 19; Q₃ = 51; IQR = 32
  3. 132 12!/10! = 132
  4. 2 μ = 43; (4 + 1 + 0 + 1 + 4)/5 = 2
  5. 3/4 9/(9 + 3) = 3/4
  6. 5 5!/(4! × 1!) = 5
  7. 111/7 (19 × 10 + 8 × 4)/(10 + 4) = 111/7
  8. 48 Q₁ = 38; Q₃ = 86; IQR = 48
  9. 240240 14!/9! = 240240
  10. 18 μ = 31; (36 + 9 + 0 + 9 + 36)/5 = 18
  11. 7/9 14/(14 + 4) = 7/9
  12. 91 14!/(2! × 12!) = 91
Review worksheet: 12 questions

  1. A score of 30 occurs 7 times; a score of 30 occurs 8 times. Find the mean as a fraction or decimal.

  2. Data: 36, 42, 48, 54, 60, 66, 72, 78. Using medians of the lower four and upper four values, find the interquartile range.

  3. Club A has 12 bus users and 7 walkers. Club B has 6 bus users. Given that a student is in Club A, what is the probability they use the bus? Give a simplified fraction.

  4. Find the population variance of 10, 16, 22, 28, 34. Divide by 5.

  5. How many different groups of 5 can be chosen from 14 distinct people, if order does not matter?

  6. How many ordered selections of 5 distinct items can be made from 10 items without repetition?

  7. A score of 18 occurs 2 times; a score of 28 occurs 2 times. Find the mean as a fraction or decimal.

  8. Data: 38, 45, 52, 59, 66, 73, 80, 87. Using medians of the lower four and upper four values, find the interquartile range.

  9. Club A has 13 bus users and 10 walkers. Club B has 13 bus users. Given that a student is in Club A, what is the probability they use the bus? Give a simplified fraction.

  10. Find the population variance of 10, 17, 24, 31, 38. Divide by 5.

  11. How many different groups of 5 can be chosen from 9 distinct people, if order does not matter?

  12. How many ordered selections of 4 distinct items can be made from 11 items without repetition?

Answer key
  1. 30 (30 × 7 + 30 × 8)/(7 + 8) = 30
  2. 24 Q₁ = 45; Q₃ = 69; IQR = 24
  3. 12/19 12/(12 + 7) = 12/19
  4. 72 μ = 22; (144 + 36 + 0 + 36 + 144)/5 = 72
  5. 2002 14!/(5! × 9!) = 2002
  6. 30240 10!/5! = 30240
  7. 23 (18 × 2 + 28 × 2)/(2 + 2) = 23
  8. 28 Q₁ = 48.5; Q₃ = 76.5; IQR = 28
  9. 13/23 13/(13 + 10) = 13/23
  10. 98 μ = 24; (196 + 49 + 0 + 49 + 196)/5 = 98
  11. 126 9!/(5! × 4!) = 126
  12. 7920 11!/7! = 7920

Related ready-made worksheets: Grade 12 Statistics & probability worksheet