Cavalieri’s Principle

Cavalieri’s Principle

A method, with formula given below, of finding the volume of any solid for which cross-sections by parallel planes have equal areas. This includes, but is not limited to, cylinders and prisms.

Formula:

Volume = Bh, where B is the area of a cross-section and h is the height of the solid.

See also

Volume by parallel cross-sections

Key Formula

V = B \cdot h

Where:

$V$ = Volume of the solid
$B$ = Area of each cross-section taken by a plane parallel to the base
$h$ = Height of the solid (perpendicular distance between the two end planes)

Worked Example

Problem: A right cylinder and an oblique cylinder both have circular bases with radius 5 cm and a height of 12 cm. Use Cavalieri's Principle to find the volume of the oblique cylinder.

Step 1: Identify the cross-sectional area. Every horizontal slice of the right cylinder is a circle with radius 5 cm.

B = \pi r^2 = \pi(5)^2 = 25\pi \text{ cm}^2

Step 2: Apply Cavalieri's Principle. The oblique cylinder is tilted, but at every height its cross-section is still a circle with the same radius 5 cm. So every cross-sectional area equals that of the right cylinder.

B_{\text{oblique}} = 25\pi \text{ cm}^2 = B_{\text{right}}

Step 3: Since both solids have the same cross-sectional area at every height and the same perpendicular height, Cavalieri's Principle guarantees they have equal volumes.

V = B \cdot h = 25\pi \cdot 12 = 300\pi \text{ cm}^3

Step 4: Compute the numerical value.

V = 300\pi \approx 942.5 \text{ cm}^3

Answer: The oblique cylinder has a volume of

300\pi \approx 942.5

cm³, the same as the right cylinder.

Another Example

This example uses a non-circular cross-section (a regular hexagon) to show that Cavalieri's Principle works for any shape where all parallel cross-sections have equal area, not just circles.

Problem: A hexagonal prism has a regular hexagonal base with side length 4 cm and a height of 10 cm. Use Cavalieri's Principle to find its volume.

Step 1: Every cross-section of the prism parallel to the base is an identical regular hexagon with side length 4 cm. Cavalieri's Principle applies because all cross-sections have the same area.

Step 2: Find the area of a regular hexagon with side length

s = 4

cm using the hexagon area formula.

B = \frac{3\sqrt{3}}{2}s^2 = \frac{3\sqrt{3}}{2}(4)^2 = \frac{3\sqrt{3}}{2} \cdot 16 = 24\sqrt{3} \text{ cm}^2

Step 3: Apply the volume formula

V = Bh

V = 24\sqrt{3} \cdot 10 = 240\sqrt{3} \text{ cm}^3

Step 4: Approximate the result.

V \approx 240 \times 1.732 = 415.7 \text{ cm}^3

Answer: The hexagonal prism has a volume of

240\sqrt{3} \approx 415.7

cm³.

Frequently Asked Questions

Does Cavalieri's Principle work for oblique (tilted) solids?

Yes. Cavalieri's Principle is especially useful for oblique solids. If an oblique solid and a corresponding right solid have the same cross-sectional area at every height, they share the same volume. This is why an oblique cylinder has the same volume formula as a right cylinder with the same base and perpendicular height.

What is the difference between Cavalieri's Principle and the disk/washer method in calculus?

Cavalieri's Principle is a geometric comparison tool: it tells you two solids have the same volume if their cross-sections match at every level. The disk/washer method is a calculus technique that integrates cross-sectional areas that change from slice to slice. You can think of the disk method as a generalization — when the cross-sectional area is constant, the integral reduces to

V = Bh

, which is exactly Cavalieri's formula.

Can Cavalieri's Principle be used in two dimensions for area?

Yes. A 2D version of Cavalieri's Principle states that if two plane regions have equal-length cross-sections at every height, they have equal areas. For example, a parallelogram and a rectangle with the same base and height have the same area because horizontal slices cut equal line segments through each shape.

Cavalieri's Principle (V = Bh) vs. Volume by Parallel Cross Sections (Integration)

	Cavalieri's Principle (V = Bh)	Volume by Parallel Cross Sections (Integration)
Definition	Solids with constant cross-sectional area have volume = area × height	Volume found by integrating varying cross-sectional areas along an axis
Formula	$V = Bh$	$V = \int_a^b A(x)\,dx$
Cross-section requirement	All cross-sections must have the same area	Cross-sectional area can change at each height
When to use	Prisms, cylinders, or comparing a tilted solid to a right solid	Cones, spheres, or any solid with a varying cross-section
Math level	Geometry (no calculus needed)	Requires integral calculus

Why It Matters

Cavalieri's Principle appears in geometry courses when you need to justify why oblique prisms and oblique cylinders share the same volume formula as their right counterparts. It also provides the conceptual foundation for the integral-based volume methods you encounter in AP Calculus. Understanding it helps you see volume not as a memorized formula but as the accumulation of cross-sectional slices — a perspective that carries through much of higher mathematics.

Common Mistakes

Mistake: Using slant height instead of perpendicular height in the formula

V = Bh

Correction: The

h

in Cavalieri's Principle is always the perpendicular distance between the two parallel planes bounding the solid. For an oblique cylinder or prism, measure the height straight up, not along the tilted side.

Mistake: Applying

V = Bh

to solids whose cross-sectional area changes with height, such as cones or pyramids.

Correction: Cavalieri's formula

V = Bh

requires that every cross-section has the same area

B

. A cone's cross-sections shrink as you move toward the apex, so you need a different formula (

V = \frac{1}{3}Bh

) or integration.

Related Terms

Volume — The quantity Cavalieri's Principle calculates
Volume by Parallel Cross Sections — Generalized integration method for varying cross-sections
Cylinder — Common solid where the principle applies directly
Prism — Another solid with constant cross-sectional area
Parallel Planes — The slicing planes used in the principle
Solid Geometry — Branch of geometry where this principle is used
Formula — V = Bh is the key formula derived from the principle