Volume of Solids

Finding volumes of prisms, pyramids, cylinders, cones, and spheres

Volume of Solids

Prisms

Volume: $V = Bh$

where $B$ = area of base, $h$ = height

Rectangular prism (box): $V = lwh$

Cube: $V = s^3$

Cylinder

$V = \pi r^2 h$

where $r$ = radius, $h$ = height

Pyramids

Volume: $V = \frac{1}{3}Bh$

where $B$ = area of base, $h$ = height

Note: Pyramids are $\frac{1}{3}$ the volume of a prism with the same base and height!

Cone

$V = \frac{1}{3}\pi r^2 h$

where $r$ = radius of base, $h$ = height

Note: Cones are $\frac{1}{3}$ the volume of a cylinder with the same base and height!

Sphere

$V = \frac{4}{3}\pi r^3$

where $r$ = radius

Key Patterns

Prisms and cylinders: Full base times height
Pyramids and cones: $\frac{1}{3}$ base times height
Sphere: Use $\frac{4}{3}\pi r^3$

📚 Practice Problems

1Problem 1easy

❓ Question:

Find the volume of a rectangular prism with length 8 cm, width 5 cm, and height 3 cm.

💡 Show Solution

Step 1: Recall the volume formula: Volume = length × width × height

Step 2: Substitute the values: V = 8 × 5 × 3 V = 40 × 3 V = 120 cm³

Answer: The volume is 120 cm³

2Problem 2easy

❓ Question:

Find the volume of a rectangular prism with length 5, width 3, and height 4.

💡 Show Solution

Use $V = lwh$ :

$V = (5)(3)(4)$

$V = 60$

Answer: 60 cubic units

3Problem 3easy

❓ Question:

A cylinder has a radius of 4 cm and a height of 10 cm. Find its volume.

💡 Show Solution

Step 1: Recall the cylinder volume formula: V = πr²h

Step 2: Identify the values: Radius r = 4 cm Height h = 10 cm

Step 3: Substitute: V = π(4)²(10) V = π(16)(10) V = 160π cm³

Step 4: Approximate (optional): V ≈ 160 × 3.14159 ≈ 502.65 cm³

Answer: Volume = 160π cm³ (≈ 502.65 cm³)

4Problem 4medium

❓ Question:

A cylinder has radius 4 and height 10. Find the volume.

💡 Show Solution

Use $V = \pi r^2 h$ :

$V = \pi(4)^2(10)$

$V = \pi(16)(10)$

$V = 160\pi$

Answer: $160\pi$ (or approximately 502.7) cubic units

5Problem 5medium

❓ Question:

Find the volume of a cone with radius 6 m and height 8 m.

💡 Show Solution

Step 1: Recall the cone volume formula: V = (1/3)πr²h

Step 2: Identify the values: Radius r = 6 m Height h = 8 m

Step 3: Substitute: V = (1/3)π(6)²(8) V = (1/3)π(36)(8) V = (1/3)(288π) V = 96π m³

Step 4: Approximate: V ≈ 96 × 3.14159 ≈ 301.59 m³

Step 5: Note: Cone volume is 1/3 of cylinder volume with same base and height

Answer: Volume = 96π m³ (≈ 301.59 m³)

6Problem 6medium

❓ Question:

A sphere has a radius of 9 cm. Find its volume.

💡 Show Solution

Step 1: Recall the sphere volume formula: V = (4/3)πr³

Step 2: Substitute r = 9: V = (4/3)π(9)³ V = (4/3)π(729) V = (4 × 729π)/3 V = 2916π/3 V = 972π cm³

Step 3: Approximate: V ≈ 972 × 3.14159 ≈ 3053.63 cm³

Answer: Volume = 972π cm³ (≈ 3053.63 cm³)

7Problem 7hard

❓ Question:

A cone and a cylinder have the same radius of 6 and the same height of 9. How many times greater is the volume of the cylinder than the cone?

💡 Show Solution

Cylinder volume: $V_{cyl} = \pi r^2 h = \pi(6)^2(9) = 324\pi$

Cone volume: $V_{cone} = \frac{1}{3}\pi r^2 h = \frac{1}{3}\pi(6)^2(9) = 108\pi$

Ratio: $\frac{V_{cyl}}{V_{cone}} = \frac{324\pi}{108\pi} = 3$

Answer: The cylinder's volume is 3 times the cone's volume

(This is always true for cone and cylinder with same base and height!)

8Problem 8hard

❓ Question:

A rectangular swimming pool is 25 m long, 10 m wide, and has an average depth of 2 m. If water costs $3 per cubic meter, how much does it cost to fill the pool?

💡 Show Solution

Step 1: Find the volume of the pool: V = length × width × depth V = 25 × 10 × 2 V = 500 m³

Step 2: Calculate the cost: Cost = Volume × Price per m³ Cost = 500 × $3 Cost =$ 1500

Step 3: Understand the context: The pool holds 500 cubic meters of water At $3 per cubic meter, total cost is$ 1500

Answer: It costs $1500 to fill the pool

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