Volume of Cylinders, Cones, and Spheres

Understanding the volume of curved 3D shapes opens up a world of real-world applications! From soda cans to ice cream cones to basketballs, these formulas help us calculate capacity and make practical decisions.

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Review: What Is Volume?

Volume measures the amount of space inside a 3D shape.

Think of it as:

• How much liquid it can hold
• How many unit cubes fit inside
• The capacity of the shape

Units: Cubic units (cm³, in³, m³, ft³)

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Volume of a Cylinder

A cylinder has two circular bases and a curved side (like a soda can).

Formula:

V = πr²h

Where:

• r = radius of the circular base
• h = height of the cylinder
• π ≈ 3.14 or use calculator's π button

Think of it as: Area of base × height

• Base area = πr²
• Multiply by height: πr² × h = πr²h

Example 1: Find the volume of a cylinder with radius 3 cm and height 10 cm. Use π ≈ 3.14.

Solution: V = πr²h V = π × (3)² × 10 V = π × 9 × 10 V = 90π V ≈ 90 × 3.14 V ≈ 282.6 cm³

Answer: 90π cm³ or approximately 282.6 cm³

Example 2: A cylindrical water tank has diameter 8 ft and height 12 ft. What's the volume?

Solution: Diameter = 8 ft, so radius = 4 ft

V = π(4)²(12) V = π × 16 × 12 V = 192π V ≈ 603.2 ft³

Answer: 192π ft³ or about 603 ft³

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Volume of a Cone

A cone has one circular base and comes to a point (apex) at the top.

Formula:

V = (1/3)πr²h

Where:

• r = radius of the circular base
• h = height (perpendicular from base to apex)
• π ≈ 3.14

Key insight: A cone's volume is 1/3 of a cylinder with the same base and height!

Example 1: Find the volume of a cone with radius 5 in and height 9 in.

Solution: V = (1/3)πr²h V = (1/3)π(5)²(9) V = (1/3)π(25)(9) V = (1/3)π(225) V = 75π V ≈ 235.5 in³

Answer: 75π in³ or about 235.5 in³

Example 2: An ice cream cone has radius 2 cm and height 12 cm. How much ice cream fits inside?

Solution: V = (1/3)π(2)²(12) V = (1/3)π(4)(12) V = (1/3)π(48) V = 16π V ≈ 50.24 cm³

Answer: 16π cm³ or about 50 cm³

Example 3: Compare volumes: Cylinder vs. Cone (same base and height)

Cylinder: r = 3, h = 6 V = π(3)²(6) = 54π

Cone: r = 3, h = 6 V = (1/3)π(3)²(6) = 18π

Cone volume = (1/3) × Cylinder volume ✓

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Volume of a Sphere

A sphere is perfectly round in all directions (like a ball).

Formula:

V = (4/3)πr³

Where:

• r = radius of the sphere
• π ≈ 3.14

Example 1: Find the volume of a sphere with radius 6 cm.

Solution: V = (4/3)πr³ V = (4/3)π(6)³ V = (4/3)π(216) V = (4/3) × 216π V = 288π V ≈ 904.3 cm³

Answer: 288π cm³ or about 904 cm³

Example 2: A basketball has diameter 24 cm. What's its volume?

Solution: Diameter = 24 cm, so radius = 12 cm

V = (4/3)π(12)³ V = (4/3)π(1,728) V = (4/3) × 1,728π V = 2,304π V ≈ 7,238.2 cm³

Answer: 2,304π cm³ or about 7,238 cm³

Example 3: A spherical water droplet has radius 0.5 mm. Find its volume.

Solution: V = (4/3)π(0.5)³ V = (4/3)π(0.125) V = (0.5/3)π V ≈ 0.524 mm³

Answer: About 0.52 mm³

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Comparing the Three Formulas

Cylinder: V = πr²h

• "Pi r squared h"
• Base area × height

Cone: V = (1/3)πr²h

• "One-third pi r squared h"
• 1/3 of cylinder volume

Sphere: V = (4/3)πr³

• "Four-thirds pi r cubed"
• Only depends on radius

Pattern: All use π because they involve circles!

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Finding Unknown Dimensions

Sometimes you know the volume and need to find a dimension.

Example 1: A cylinder has volume 200π cm³ and height 8 cm. Find the radius.

Solution: V = πr²h 200π = πr²(8) 200π = 8πr²

Divide by 8π: 25 = r² r = 5 cm

Answer: r = 5 cm

Example 2: A sphere has volume 288π in³. Find the radius.

Solution: V = (4/3)πr³ 288π = (4/3)πr³

Multiply by 3/4: (3/4) × 288π = πr³ 216π = πr³

Divide by π: 216 = r³ r = 6 in

Answer: r = 6 in

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Real-World Applications

Cylinders:

Food cans: Calculate how much soup or soda fits

• Soda can: r ≈ 3 cm, h ≈ 12 cm
• V ≈ 339 cm³ (about 339 mL)

Water tanks: Storage capacity

• Tank: r = 5 ft, h = 10 ft
• V ≈ 785 ft³

Pipes: How much water flows through

• Pipe: r = 2 in, length = 100 in
• V ≈ 1,256 in³

Cones:

Ice cream cones: How much ice cream fits

• Waffle cone: r = 3 cm, h = 10 cm
• V ≈ 94 cm³

Traffic cones: Volume of material

• Orange cone: r = 6 in, h = 18 in
• V ≈ 678 in³

Funnels: Liquid capacity

Spheres:

Sports balls: Air volume inside

• Basketball: r ≈ 12 cm
• V ≈ 7,238 cm³

Planets: Volume of celestial bodies

• Earth radius ≈ 6,371 km
• V ≈ 1.08 × 10¹² km³

Ball bearings: Industrial applications

Water balloons: How much water they hold

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Composite Shapes

Sometimes shapes are combined!

Example: A shape consists of a cylinder topped with a hemisphere (half-sphere). The radius is 4 cm and the cylinder height is 10 cm. Find total volume.

Solution:

Cylinder part: V₁ = πr²h = π(4)²(10) = 160π cm³

Hemisphere part: Full sphere: V = (4/3)π(4)³ = (256/3)π Half sphere: V₂ = (128/3)π cm³

Total: V = 160π + (128/3)π V = (480/3)π + (128/3)π V = (608/3)π V ≈ 637 cm³

Answer: (608/3)π cm³ or about 637 cm³

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Units and Conversions

Volume units must be CUBIC:

Length units → Volume units:

• cm → cm³
• m → m³
• ft → ft³
• in → in³

Important conversions:

• 1 m³ = 1,000,000 cm³
• 1 ft³ = 1,728 in³
• 1 cm³ = 1 mL (milliliter)
• 1 m³ = 1,000 liters

Example: A cylinder has r = 10 cm and h = 20 cm. Find volume in liters.

Solution: V = π(10)²(20) = 2,000π ≈ 6,283 cm³

Convert to liters: 6,283 cm³ = 6,283 mL = 6.283 liters

Answer: About 6.3 liters

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Common Mistakes to Avoid

❌ Mistake 1: Using diameter instead of radius

• Wrong: V = π(10)²h when diameter = 10
• Right: r = 5, so V = π(5)²h

❌ Mistake 2: Forgetting the 1/3 for cones

• Wrong: Cone V = πr²h
• Right: Cone V = (1/3)πr²h

❌ Mistake 3: Forgetting the 4/3 for spheres

• Wrong: V = πr³
• Right: V = (4/3)πr³

❌ Mistake 4: Wrong units (square instead of cubic)

• Wrong: Volume = 50 cm²
• Right: Volume = 50 cm³

❌ Mistake 5: Calculation errors with π

• Use calculator's π button for accuracy
• Or use 3.14 as approximation

❌ Mistake 6: Confusing height with slant height (cones)

• Use perpendicular height, not slant

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Problem-Solving Strategy

Step 1: Identify the shape

• Cylinder? Cone? Sphere? Combination?

Step 2: Write the appropriate formula

Step 3: Identify given information

• Radius or diameter?
• Height?
• Convert diameter to radius if needed

Step 4: Substitute into formula

Step 5: Calculate step-by-step

• Handle fractions carefully
• Keep π in answer or use 3.14

Step 6: Include proper units (cubic!)

Step 7: Round appropriately if needed

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Quick Reference

Cylinder: V = πr²h

Cone: V = (1/3)πr²h

Sphere: V = (4/3)πr³

Remember:

• All use radius (r), not diameter
• Cone = 1/3 of cylinder
• Units must be cubic (cm³, m³, etc.)
• Diameter = 2 × radius

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Practice Tips

Tip 1: Draw and label

• Sketch the shape
• Mark radius, height, diameter
• Check which you're given

Tip 2: Double-check formulas

• Cylinder has h
• Cone has 1/3
• Sphere has 4/3 and r³

Tip 3: Watch for diameter vs. radius

• Always convert diameter to radius!
• r = d/2

Tip 4: Use calculator wisely

• π button for best accuracy
• Or use 3.14 for approximation

Tip 5: Check reasonableness

• Cone should be 1/3 of cylinder
• Larger radius = much larger volume (squared or cubed!)

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Summary

Volume formulas for curved shapes:

Cylinder: V = πr²h

• Two circular bases
• Straight sides
• Applications: cans, tanks, pipes

Cone: V = (1/3)πr²h

• One circular base
• Comes to a point
• 1/3 of cylinder volume
• Applications: ice cream cones, funnels

Sphere: V = (4/3)πr³

• Perfectly round
• Only depends on radius
• Applications: balls, planets, bubbles

Key skills:

• Identify the correct formula
• Convert diameter to radius
• Calculate accurately with π
• Use proper cubic units

These formulas are essential for engineering, manufacturing, science, and countless real-world applications!