Cavalieri's Principle and Sphere Volume

Lesson 2 of 2: Geometric Measurement and Dimension

In this lesson:

• Understand Cavalieri's principle — equal cross-sections → equal volumes
• Derive using a hemisphere comparison
• Discover where the comes from — it's not arbitrary

Learning Objectives

By the end of this lesson, you will be able to:

1. State Cavalieri's principle and explain why equal cross-sectional areas imply equal volumes
2. Describe the hemisphere vs. cylinder-minus-cone comparison setup
3. Compute cross-sectional areas of both solids at height and show they are equal
4. Derive using Cavalieri's principle
5. Apply Cavalieri's principle informally to compare other solids

You Know the Formula — But Why ?

You already know these volume formulas:

• Cylinder:
• Cone:
• Sphere:

Today's question: Where does the come from?

Answer: it's hidden in the difference between a cylinder and a cone.

Cavalieri's Principle

If two solids have:

• The same height, and
• Equal cross-sectional areas at every height

Then: the two solids have equal volumes

The cross-sections don't need to be the same shape — only the same area

Same Volume — Different Arrangement

Shifting the coins doesn't change volume — every coin has the same area at every height

Key Clarification: Area, Not Shape

Cavalieri's principle requires equal area — not equal shape

Different shapes, same area → same contribution to volume

Quick Check: Cavalieri's Principle

Two solids have the same height. At every level from base to top, their cross-sections have equal area — but different shapes.

What can you conclude?

Think for a moment before the next slide...

Answer: Equal Volumes

✓ The two solids have equal volumes

This follows directly from Cavalieri's principle:

• Same height ✓
• Equal cross-sectional areas at every level ✓
• Therefore: equal volumes ✓

The shapes of the cross-sections are irrelevant — only the areas matter

The Plan: Hemisphere vs. Comparison Solid

Goal: Find using Cavalieri's principle

Strategy:

1. Compare a hemisphere of radius to a solid we already know
2. Show equal cross-sections at every height
3. Conclude equal volumes → compute

The comparison solid: a cylinder of radius , height , with a cone removed from inside

Two Solids to Compare

Both solids have height and base radius

Hemisphere Setup

Hemisphere of radius :

• Flat base down, at height : circle of radius
• At height : tapers to a single point
• Cross-section at any height: a circle (disk)

We need to find the radius of that circle as a function of

Cylinder-Minus-Cone Setup

Cylinder minus cone (both radius , height ):

• Cylinder cross-section: always a full circle of radius
• Cone cross-section at height : circle of radius

Watch out: the cone's apex is at the bottom (), opening upward

At height , the cone's radius equals — not

Hemisphere Cross-Section at Height

At height : use the Pythagorean Theorem on the right triangle

Computing

Pythagorean Theorem on the right triangle:

Cross-sectional area of the hemisphere at height :

Check: at , ✓   at , ✓

Quick Check: Cross-Section at

A hemisphere has radius . Find the cross-sectional area at height .

Use:

Try the calculation before advancing

Answer:

This is ¾ of the base area — makes sense: halfway up, still a large cross-section

Cylinder-Minus-Cone Cross-Section at Height

At height : an annulus (ring) with outer radius , inner radius

Why the Cone's Radius at Height Equals

The cone has apex at the bottom () and base radius at height :

• Radius grows linearly: from 0 to as goes from 0 to
• Rate of growth: , so radius at height is exactly

This only works because height = radius =

The cone has slope 1 — a 45° half-angle

Computing

Annulus area = (cylinder area) − (cone area)

Compare to hemisphere:

Areas Match at Every Height

By Cavalieri's principle: equal areas at every height → equal volumes

Transition: From Areas to Volume

What we've established:

(because their cross-sectional areas are equal at every height)

What we already know:

Now subtract — and then double

Step 1: Volume of Cylinder and Cone

Both solids have radius and height :

These formulas come from HSG.GMD.A.1 — the previous lesson

Step 2: Hemisphere Volume by Subtraction

Step 3: Double for the Full Sphere

A sphere consists of two hemispheres:

Why ? The Arithmetic Explained

The fraction is not mysterious — trace the arithmetic:

• The 1 comes from the cylinder:
• The comes from the cone:
• The 2 comes from doubling the hemisphere

Quick Check: Sphere Volume

A sphere has radius 3 cm. Find the volume. Leave in terms of .

Use:

Try the calculation before advancing

Answer: cm³

Approximately cm³

Key step: Cube first (), then multiply by

Key Takeaways

✓ Cavalieri's principle: equal cross-section areas at every height → equal volumes

✓ Hemisphere = cylinder − cone (by Cavalieri's principle, since areas match)

✓ Sphere volume: — derived from cylinder and cone formulas

✓ The = — not arbitrary, pure arithmetic

Cone opens upward — apex at bottom, radius at height equals

Cavalieri needs equal areas, not equal shapes — disk and annulus can match

Hemisphere volume is — double it for the full sphere

Hemisphere radius at height : use Pythagorean Theorem: , not

Historical Note: Archimedes' Discovery

Archimedes (287–212 BCE) discovered this argument first — over 2,000 years ago

• He used a different but equivalent method (lever arguments)
• Cavalieri formalized the cross-section principle in the 17th century
• This argument is one of the oldest results in what we now call integral calculus

The same idea — accumulating cross-sections — becomes the disk method in calculus

What's Next

HSG.GMD.A.3: Using to solve real-world problems

• Volumes of spherical tanks, Earth and Moon models, ball bearings
• Comparing volumes of spheres and cylinders
• Multi-step problems combining sphere, cylinder, and cone volumes

Bigger picture: This Cavalieri's principle idea — comparing cross-sections — becomes integration in Calculus. Today you saw informal calculus in action.