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Geometry

Volume

Lesson

Volume measures the 3D space inside a solid. The formulas form natural pairs: prisms/cylinders use V=BhV = B h (base times height), and cones/pyramids are one-third of that.

Formulas

  • Cube: s3s^3
  • Rect prism: ℓwh\ell w h
  • Cylinder: πr2h\pi r^2 h
  • Cone: 13πr2h\tfrac{1}{3} \pi r^2 h
  • Pyramid: 13Bh\tfrac{1}{3} B h (B = base area)
  • Sphere: 43πr3\tfrac{4}{3} \pi r^3

Use π≈3.14\pi \approx 3.14 for problems in this topic.

Worked example

Cylinder with r=3, h=4:

V=3.14⋅9⋅4=113.04V = 3.14 \cdot 9 \cdot 4 = 113.04

Practice

Work through these. Stuck? Click Get a hint.

Warm-Up

Quick problems to get going.

Problem 1

Cube s=3\text{Cube } s = 3

Problem 2

Rect prism 2×3×4\text{Rect prism } 2 \times 3 \times 4

Problem 3

Cylinder r=2,h=5\text{Cylinder } r = 2, h = 5

Problem 4

Sphere r=3\text{Sphere } r = 3

Practice

Standard problems matching the lesson.

Problem 5

Cube s=5\text{Cube } s = 5

Problem 6

Cube s=10\text{Cube } s = 10

Problem 7

Rect prism 4×5×6\text{Rect prism } 4 \times 5 \times 6

Problem 8

Rect prism 2×2×10\text{Rect prism } 2 \times 2 \times 10

Problem 9

Cylinder r=3,h=4\text{Cylinder } r=3, h=4

Problem 10

Cylinder r=5,h=2\text{Cylinder } r=5, h=2

Problem 11

Cylinder r=1,h=10\text{Cylinder } r=1, h=10

Problem 12

Sphere r=6\text{Sphere } r=6

Problem 13

Cone r=3,h=4\text{Cone } r=3, h=4

Problem 14

Cone r=6,h=5\text{Cone } r=6, h=5

Problem 15

Square pyramid base 4, h=6\text{Square pyramid base 4, } h=6

Problem 16

Square pyramid base 5, h=12\text{Square pyramid base 5, } h=12

Problem 17

Cube s=7\text{Cube } s=7

Problem 18

Box 10×5×3\text{Box } 10 \times 5 \times 3

Challenge

Harder problems — edge cases, trickier numbers, multiple steps.

Problem 19

Cube V=64. Side?\text{Cube V=64. Side?}

Problem 20

Cube V=729. Side?\text{Cube V=729. Side?}

Problem 21

Cylinder V=565.2, r=6. Height?\text{Cylinder V=565.2, r=6. Height?}

Problem 22

Cone r=4, V=100.48. Height?\text{Cone r=4, V=100.48. Height?}

Problem 23

Box V=120, base 3×4. Height?\text{Box V=120, base } 3 \times 4. \text{ Height?}

Problem 24

Cylinder r=10,h=10\text{Cylinder } r=10, h=10

Problem 25

Hemisphere r=3\text{Hemisphere } r=3

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