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 5, width 3, and height 4.

💡 Show Solution

Use $V = lwh$ :

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

$V = 60$

Answer: 60 cubic units

2Problem 2medium

❓ 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

3Problem 3hard

❓ 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!)

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